(original) (raw)

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(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)f(.)h(.)g(.)128 b(23)-34 2795 y(2.2)100 b(F)-8 b(ourier)31 b(transform)h(o)m(v)m(er)h Fq(T)1247 2759 y Fp(n)1383 2795 y Fw(.)50 b(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)128 b(24)-34 2915 y(2.3)100 b(Expander)34 b(Graphs)e(Construction)85 b(.)50 b(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.) g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)128 b(25)-34 3036 y(2.4)100 b(The)33 b(Ra)m(yleigh)f(quotien)m(t)g(of)g(Op)s (erators)53 b(.)d(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)128 b(27)-34 3156 y(2.5)100 b(Lattice)32 b(Dualit)m(y:)41 b(Banaszczyk's)36 b(T)-8 b(ransference)34 b(Theorem)87 b(.)50 b(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.) 128 b(30)-34 3276 y(2.6)100 b(Gaussian-lik)m(e)31 b(Measures)j(on)f (Lattices)79 b(.)50 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m(elopmen)m(t)g(of)e(mathematics.)-180 1978 y(Diric)m(hlet,)30 b(in)i(his)g(w)m(ork)h(dealing)e(with)g(the)i(con)m(v)m(ergence)i(of)d (F)-8 b(ourier)31 b(series,)i(de\014ned)h(the)e(notion)g(of)f(function) -180 2098 y(as)f(w)m(e)h(kno)m(w)f(it)f(to)s(da)m(y)-8 b(.)42 b(Riemann)29 b(in)m(tro)s(duced)h(his)f(notion)g(of)g(in)m (tegral)f(in)h(his)g(w)m(ork)i(on)e(trigonometric)e(series.)-180 2219 y(Can)m(tor's)32 b(study)h(of)e(the)h(so-called)e(sets)j(of)e (uniqueness)i(led)e(him)f(to)h(the)h(dev)m(elopmen)m(t)h(of)e(the)h (theory)g(of)f(sets.)-180 2339 y(T)-8 b(o)s(da)m(y)42 b(harmonic)e(analysis)g(is)h(used)i(in)d(ev)m(ery)j(branc)m(h)g(of)d (mathematics)g(including)g(group)h(theory)h(\(where)-180 2459 y(it)31 b(w)m(as)j(started)f(b)m(y)g(F)-8 b(rob)s(enius\),)32 b(probabilit)m(y)e(theory)-8 b(,)34 b(com)m(binatorics,)d(di\013eren)m (tial)f(equations,)j(and)g(n)m(um)m(b)s(er)-180 2580 y(theory)-8 b(.)-80 2700 y(Metho)s(ds)28 b(of)f(harmonic)f(analysis)g (also)h(found)g(their)g(w)m(a)m(y)h(to)f(Computer)h(Science.)42 b(One)28 b(of)f(the)h(most)e(imp)s(or-)-180 2820 y(tan)m(t)g (applications)d(in)i(Computer)g(Science)h(is)f(the)h(F)-8 b(ast)25 b(F)-8 b(ourier)25 b(T)-8 b(ransform)25 b(\(FFT\))g(algorithm) d(of)j(Co)s(oley)g(and)-180 2941 y(T)-8 b(uk)m(ey)34 b([CT65])e(and)g(its)f(application)f(to)h(the)h(fast)g(m)m (ultiplication)27 b(of)32 b(n)m(um)m(b)s(ers)g(b)m(y)h(Sc)m(h\177)-49 b(onhage)33 b(and)f(Strassen)-180 3061 y([SS71].)41 b(Ch)m(ung,)28 b(Diaconis)c(and)h(Graham)f(used)i(the)g(con)m(v)m(olution)f Ft($)g Fw(m)m(ultiplication)20 b(prop)s(ert)m(y)27 b(of)d(the)i(F)-8 b(ourier)-180 3182 y(transform)36 b(to)h(analyze)g(random)g(w)m(alks)h (on)f(graphs)g([CDG87)o(].)58 b(Linial,)36 b(Mansour)i(and)f(Nisan)g ([LMN93])g(in-)-180 3302 y(v)m(estigated)c(the)g(prop)s(erties)f(of)g (the)g(F)-8 b(ourier)31 b(co)s(e\016cien)m(ts)j(of)e(functions)g (computed)g(b)m(y)i(the)e Fs(AC)3449 3266 y Fr(0)3521 3302 y Fw(circuits)g(and)-180 3422 y(obtained)38 b(result)h(ab)s(out)f (the)i(learnabilit)m(y)c(of)i Fs(AC)1753 3386 y Fr(0)1831 3422 y Fw(functions.)62 b(The)40 b(KM)f(learning)e(algorithm)f(of)i (Kushile-)-180 3543 y(vitz)g(and)h(Mansour)h([KM93)o(])f(is)f(based)i (on)f(estimating)d(the)j(F)-8 b(ourier)38 b(co)s(e\016cien)m(ts)i(of)e (the)h(function)f(learned.)-180 3663 y(F)-8 b(or)38 b(more)g (applications)f(of)i(harmonic)e(analysis)i(in)f(learning)f(theory)j (see)g([Jac95,)f(BFJ)3219 3627 y Fr(+)3278 3663 y Fw(94)o(].)63 b(Kahn,)40 b(Kalai)-180 3784 y(and)33 b(Linial)e([KKL88)o(])i(used)h (the)g(F)-8 b(ourier)32 b(transform)g(and)h(Bec)m(kner's)j(Lemmas)c ([Bec75)q(])h(to)g(sho)m(w)h(that)f(ev)m(ery)-180 3904 y(balanced)e(Bo)s(olean)f(function)g(has)h(a)g(v)-5 b(ariable)29 b(with)i(large)f(in\015uence.)44 b(Th)m(\023)-46 b(erien)31 b([Th)m(\023)-46 b(e94)q(])31 b(applied)f(generalized)-180 4024 y(harmonic)k(analysis)h(to)g(study)i(of)d(circuits)h(with)g(MOD) 1938 4039 y Fp(m)2040 4024 y Fw(gates.)52 b(Gabb)s(er)35 b(and)h(Galil)c(successfully)37 b(analyzed)-180 4145 y(a)32 b(mo)s(di\014ed)f(construction)h(of)f(Margulis)g([Mar73])h (using)g(harmonic)e(analysis)i(on)g(the)g(t)m(w)m(o)h(dimensional)d (torus)-180 4265 y Fq(T)-117 4229 y Fr(2)-74 4265 y Fw(.)42 b(Banaszczyk)30 b(sho)m(w)m(ed)h(a)d(transference)h(theorem)f(in)g (lattices)f(using)h(F)-8 b(ourier)27 b(transforms)h(of)g(Gaussian-lik)m (e)-180 4385 y(measures)j(in)f Fq(R)414 4349 y Fp(n)467 4385 y Fw(.)43 b(J.)30 b(Naor)g(and)h(M.)g(Naor)f([NN93])g(used)i(the)f (F)-8 b(ourier)29 b(transform)g(to)h(design)h(p)s(olynomial)c(size)-180 4506 y(sample)32 b(spaces)i(of)e Fs(")p Fw(-biased)g(\(log)16 b Fs(n)p Fw(\)-wise)33 b(indep)s(enden)m(t)h(random)d(v)-5 b(ariables.)1896 5649 y(3)p eop %%Page: 4 8 4 7 bop eop %%Page: 5 9 5 8 bop -180 1065 a Fy(Chapter)78 b(1)-180 1482 y(Harmonic)f(Analysis)e (o)-6 b(v)g(er)76 b(Finite)h(Ab)6 b(elian)-180 1731 y(Groups)-180 2248 y Fn(1.1)161 b(In)l(tro)t(duction)-180 2470 y Fw(Let)30 b Fs(A)g Fw(b)s(e)g(a)g(measure)g(space)h(with)f(a)f(non-negativ)m(e)h (measure)g Fs(\026)g Fw(and)g(the)g(corresp)s(onding)g(Leb)s(esgue)h (in)m(tegral.)-180 2590 y(F)-8 b(or)32 b(complex)g(v)-5 b(alued)32 b(functions)h Fs(f)5 b(;)17 b(g)31 b Ft(2)d Fq(C)1439 2554 y Fp(A)1534 2590 y Fw(w)m(e)34 b(de\014ne)-35 2802 y Ft(\017)49 b Fu(p)s(oin)m(t)m(wise)36 b(m)m(ultiplication)28 b Fw(\()p Fs(f)11 b(g)t Fw(\)\()p Fs(x)p Fw(\))26 b(=)i Fs(f)11 b Fw(\()p Fs(x)p Fw(\))p Fs(g)t Fw(\()p Fs(x)p Fw(\))p Fs(;)49 b(x)28 b Ft(2)g Fs(A)p Fw(,)-35 3014 y Ft(\017)49 b Fu(inner)37 b(pro)s(duct)32 b Ft(h)p Fs(f)5 b(;)17 b(g)t Ft(i)27 b Fw(=)1127 2934 y Fm(R)1210 3014 y Fs(f)11 b Fw(\()p Fs(y)t Fw(\))p 1397 2927 178 4 v Fs(g)t Fw(\()p Fs(y)t Fw(\))j Fs(d\026)p Fw(\()p Fs(y)t Fw(\),)31 b(where)p 2168 2959 56 4 v 34 w Fs(x)i Fw(is)f(the)h(complex) f(conjugate)h(of)f Fs(x)c Ft(2)g Fq(C)20 b Fw(,)-35 3211 y Ft(\017)49 b Fl(p)p Fu(-norm)32 b Ft(jj)p Fs(f)11 b Ft(jj)622 3226 y Fp(p)688 3211 y Fw(=)791 3101 y Fm(\020)868 3131 y(R)951 3211 y Ft(j)p Fs(f)g Fw(\()p Fs(y)t Fw(\))p Ft(j)1194 3175 y Fp(p)1248 3211 y Fs(d\026)p Fw(\()p Fs(y)t Fw(\))1486 3101 y Fm(\021)1544 3123 y Fr(1)p Fp(=p)1654 3211 y Fw(,)-35 3408 y Ft(\017)49 b Fu(con)m(v)m(olution)31 b Fw(\()p Fs(f)i Ft(\003)22 b Fs(g)t Fw(\)\()p Fs(x)p Fw(\))27 b(=)1201 3328 y Fm(R)1284 3408 y Fs(f)11 b Fw(\()p Fs(y)t Fw(\))p Fs(g)t Fw(\()p Fs(x)21 b Ft(\000)i Fs(y)t Fw(\))17 b Fs(d\026)p Fw(\()p Fs(y)t Fw(\).)-80 3620 y(Let)34 b Fs(G)f Fw(b)s(e)h(a)g(\014nite)f(ab)s(elian)f(group)i (written)f(additiv)m(ely)-8 b(.)46 b(Let)34 b Fs(n)c Fw(=)g Ft(j)p Fs(G)p Ft(j)p Fw(.)46 b Fu(Measure)41 b(on)e Fl(G)p Fu(:)46 b Fw(W)-8 b(e)34 b(endo)m(w)h Fs(G)-180 3741 y Fw(with)28 b(the)i(measure)f Fs(\026)p Fw(\()p Fs(g)t Fw(\))d(=)i(1)p Fs(=n;)45 b(g)31 b Ft(2)d Fs(G)p Fw(.)42 b(W)-8 b(e)29 b(de\014ne)h(the)g(inner)e(pro)s(duct,)i(the)f Fs(p)p Fw(-norm)e(and)i(the)h(con)m(v)m(olution)-180 3861 y(on)i(the)h(space)h Fq(C)450 3825 y Fp(G)548 3861 y Fw(as)f(ab)s(o)m(v)m(e.)-80 3983 y(By)g Fq(S)134 3947 y Fr(1)200 3983 y Fw(w)m(e)h(denote)f(the)g(m)m(ultiplicativ)m(e)c (group)j(of)h(complex)f(n)m(um)m(b)s(ers)h(of)f(mo)s(dulus)g(1.)-180 4213 y Fu(De\014nition)k(1.1.1)49 b Fw(A)31 b Fu(c)m(haracter)h Fs(\037)f Fw(of)f Fs(G)h Fw(is)g(a)g(homomorphism)d Fs(G)g Ft(!)f Fq(S)2678 4177 y Fr(1)2712 4213 y Fw(.)43 b(The)32 b Fu(unit)j(c)m(haracter)c(1)g Fw(is)g(the)-180 4333 y(c)m(haracter)i(whic)m(h)h(assigns)e(1)h(to)f(ev)m(ery)i Fs(a)28 b Ft(2)g Fs(G)p Fw(.)-180 4527 y(Since)33 b Fs(G)f Fw(is)g(\014nite)h(a)f(function)g Fs(\037)c Fw(:)f Fs(G)h Ft(!)f Fq(C)59 b Fw(is)32 b(a)g(c)m(haracter)i(i\013)d(it)h (satis\014es)1277 4752 y Fs(\037)p Fw(\()p Fs(a)22 b Fw(+)g Fs(b)p Fw(\))28 b(=)g Fs(\037)p Fw(\()p Fs(a)p Fw(\))p Fs(\037)p Fw(\()p Fs(b)p Fw(\))p Fs(;)50 b(a;)17 b(b)28 b Ft(2)g Fs(G:)-180 4977 y Fw(Note)33 b(that)-35 5188 y Ft(\017)49 b Fs(\037)p Fw(\()p Fs(a)p Fw(\))252 5152 y Fp(n)327 5188 y Fw(=)27 b Fs(\037)p Fw(\()p Fs(n)c Ft(\001)e Fs(a)p Fw(\))28 b(=)g Fs(\037)p Fw(\(0\))f(=)h(1.)-35 5400 y Ft(\017)49 b Fs(\037)p Fw(\()p Ft(\000)p Fs(a)p Fw(\))28 b(=)p 461 5313 189 4 v 28 w Fs(\037)p Fw(\()p Fs(a)p Fw(\).)1896 5649 y(5)p eop %%Page: 6 10 6 9 bop -360 68 a Fw(6)767 b Fk(CHAPTER)34 b(1.)76 b(HARMONIC)33 b(ANAL)-8 b(YSIS)34 b(O)m(VER)f(FINITE)g(ABELIAN)h(GR)m(OUPS)-215 400 y Ft(\017)49 b Fw(If)32 b Fs(\037)h Fw(and)g Fs( )j Fw(are)d(c)m(haracters)h(then)f Fs(\037 )k Fw(and)p 1563 345 62 4 v 32 w Fs(\037)c Fw(are)f(c)m(haracters.)-360 620 y Fu(Lemma)37 b(1.1.2)49 b Fj(Char)-5 b(acters)34 b(ar)-5 b(e)35 b(an)g(orthonormal)e(set)i(of)g(functions.)-360 806 y Fu(Pro)s(of)i(:)-360 927 y Fw(Note)c(that)f(for)g(an)m(y)h(c)m (haracter)h Fs(\037)e Fw(and)h(an)m(y)g Fs(a)28 b Ft(2)g Fs(G)899 1159 y(\037)p Fw(\()p Fs(a)p Fw(\))1104 1064 y Fm(X)1109 1276 y Fp(b)p Fo(2)p Fp(G)1264 1159 y Fs(\037)p Fw(\()p Fs(b)p Fw(\))g(=)1574 1064 y Fm(X)1580 1276 y Fp(b)p Fo(2)p Fp(G)1734 1159 y Fs(\037)p Fw(\()p Fs(a)23 b Fw(+)f Fs(b)p Fw(\))28 b(=)2215 1064 y Fm(X)2221 1276 y Fp(b)p Fo(2)p Fp(G)2376 1159 y Fs(\037)p Fw(\()p Fs(b)p Fw(\))p Fs(:)-360 1473 y Fw(If)38 b Fs(\037)f Ft(6)p Fw(=)g Fu(1)i Fw(then)f(there)h(is)f(an)g Fs(a)g Fw(suc)m(h)i(that)e Fs(\037)p Fw(\()p Fs(a)p Fw(\))f Ft(6)p Fw(=)g(1)h(and)g(hence)2205 1398 y Fm(P)2310 1502 y Fp(b)p Fo(2)p Fp(G)2463 1473 y Fs(\037)p Fw(\()p Fs(b)p Fw(\))f(=)h(0.)59 b(If)39 b(c)m(haracters)g Fs( )t(;)17 b(\037)38 b Fw(are)-360 1593 y(di\013eren)m(t)33 b(then)g Fs( )p 314 1538 V 4 w(\037)28 b Ft(6)p Fw(=)f Fu(1)33 b Fw(and)g(hence)1166 1852 y Ft(h)p Fs( )t(;)17 b(\037)p Ft(i)27 b Fw(=)1561 1785 y(1)p 1557 1829 59 4 v 1557 1921 a Fs(n)1641 1758 y Fm(X)1647 1970 y Fp(b)p Fo(2)p Fp(G)1785 1852 y Fw(\()p Fs( )p 1890 1797 62 4 v 4 w(\037)p Fw(\)\()p Fs(b)p Fw(\))h(=)g(0)p Fs(:)-360 2166 y Fw(Clearly)k Ft(h)p Fs( )t(;)17 b( )t Ft(i)27 b Fw(=)g(1.)3322 b Fi(\004)-260 2368 y Fw(Characters)33 b(with)g(the)g(p)s(oin)m(t)m(wise)f(m)m(ultiplication)c(form)k(a)g (group)2288 2343 y Fm(b)2269 2368 y Fs(G)p Fw(,)h(called)f(the)h Fu(dual)k(group)c Fw(of)f Fs(G)p Fw(.)44 b(The)-360 2488 y(unit)32 b(c)m(haracter)h(is)f(the)h(unit)f(of)873 2463 y Fm(b)854 2488 y Fs(G)p Fw(.)44 b(Ev)m(ery)34 b(\014nite)f(ab)s(elian) d(group)j(is)f(a)g(direct)h(sum)f(of)g(cyclic)g(groups.)44 b(Fix)1293 2708 y Fs(G)27 b Fw(=)h Fq(Z)1570 2723 y Fp(n)1613 2732 y Fh(1)1671 2708 y Ft(\010)23 b(\001)17 b(\001)g(\001)j(\010)j Fq(Z)2078 2723 y Fp(n)2121 2735 y Fg(k)2160 2708 y Fs(:)-360 2928 y Fw(Let)33 b Fs(!)-124 2943 y Fp(i)-69 2928 y Fw(=)28 b(exp)q(\(2)p Fs(\031)t(i=n)470 2943 y Fp(i)498 2928 y Fw(\),)33 b(a)f(primitiv)m(e)e Fs(n)1152 2943 y Fp(i)1181 2928 y Fw(-th)i(ro)s(ot)f(of)h(unit)m(y)-8 b(.)44 b(F)-8 b(or)32 b Fs(b)c Fw(=)f(\()p Fs(b)2368 2943 y Fr(1)2408 2928 y Fs(;)17 b(:)g(:)g(:)f(;)h(b)2668 2943 y Fp(k)2711 2928 y Fw(\))28 b Ft(2)g Fs(G)k Fw(let)1380 3241 y Fs(\037)1441 3256 y Fp(b)1475 3241 y Fw(\()p Fs(x)p Fw(\))c(=)1782 3117 y Fp(k)1738 3147 y Fm(Y)1744 3357 y Fp(i)p Fr(=1)1882 3241 y Fs(!)1947 3197 y Fp(b)1977 3207 y Fg(i)2003 3197 y Fp(x)2043 3207 y Fg(i)1943 3267 y Fp(i)2073 3241 y Fs(:)-360 3544 y Fw(Note)33 b(that)g Fs(\037)149 3559 y Fp(b)217 3544 y Fw(is)g(a)g(c)m(haracter)h(for)e(an)m(y)i Fs(b)c Ft(2)f Fs(G)p Fw(.)45 b(The)34 b Fs(\037)1735 3559 y Fp(b)1803 3544 y Fw(are)f(all)e(distinct.)45 b(They)35 b(are)e(all)e(c)m(haracters)k(b)s(ecause)-360 3664 y(the)e(dimension)e (of)h Fq(C)446 3628 y Fp(G)544 3664 y Fw(is)g Fs(n)h Fw(and)f(the)h(c)m(haracters)h(are)f(orthonormal.)41 b(Th)m(us)34 b(the)f(dual)f(group)g(of)g Fs(G)h Fw(is)1044 3881 y Fm(b)1025 3906 y Fs(G)27 b Fw(=)1233 3795 y Fm(n)1299 3906 y Fs(\037)1360 3921 y Fp(b)1427 3906 y Ft(j)32 b Fs(b)c Fw(=)g(\()p Fs(b)1739 3921 y Fr(1)1779 3906 y Fs(;)17 b(:)g(:)g(:)f(;)h(b)2039 3921 y Fp(k)2082 3906 y Fw(\))27 b Ft(2)h Fs(G;)2362 3795 y Fm(o)2428 3906 y Fs(:)1184 b Fw(\(1.1\))-360 4160 y Fu(Theorem)37 b(1.1.3)49 b Fj(Char)-5 b(acters)34 b(form)h(an)f(orthonormal)g(b)-5 b(asis)34 b(of)h Fq(C)2235 4124 y Fp(G)2300 4160 y Fj(.)-360 4347 y Fw(F)-8 b(rom)26 b(Theorem)i(1.1.3)f(it)g(follo)m(ws)g(that)g (ev)m(ery)j(function)d Fs(f)38 b Fw(in)27 b Fq(C)2032 4311 y Fp(G)2125 4347 y Fw(can)h(b)s(e)g(expressed)j(as)c(a)h(linear)e (com)m(bination)-360 4467 y(of)38 b(c)m(haracters.)62 b(The)39 b(co)s(e\016cien)m(t)g(of)f Fs(\037)1121 4482 y Fp(b)1193 4467 y Fw(is)g(denoted)h(b)m(y)1830 4441 y Fm(b)1812 4467 y Fs(f)11 b Fw(\()p Fs(\037)1970 4482 y Fp(b)2004 4467 y Fw(\))39 b(and)f(called)f(the)i Fu(F)-9 b(ourier)43 b(co)s(e\016cien)m(t)p Fw(.)60 b(W)-8 b(e)-360 4588 y(ha)m(v)m(e)1388 4808 y Fs(f)39 b Fw(=)1578 4713 y Fm(X)1584 4925 y Fp(b)p Fo(2)p Fp(G)1757 4781 y Fm(b)1739 4808 y Fs(f)11 b Fw(\()p Fs(\037)1897 4823 y Fp(b)1931 4808 y Fw(\))p Fs(\037)2030 4823 y Fp(b)2064 4808 y Fs(:)1548 b Fw(\(1.2\))-360 5145 y(The)37 b(function)247 5119 y Fm(b)229 5145 y Fs(f)44 b Fw(:)400 5120 y Fm(b)381 5145 y Fs(G)34 b Ft(!)f Fq(C)61 b Fw(is)36 b(called)e(the)j Fu(F)-9 b(ourier)40 b(transform)c Fw(of)f Fs(f)11 b Fw(.)53 b Fu(Measure)42 b(on)3090 5120 y Fm(b)3064 5145 y Fl(G)p Fw(:)50 b(W)-8 b(e)36 b(endo)m(w)i(the)-360 5266 y(group)-55 5241 y Fm(b)-74 5266 y Fs(G)k Fw(with)f(the)i(measure)f Fs(\027)6 b Fw(\()p Fs(g)t Fw(\))43 b(=)h(1)p Fs(;)58 b(g)47 b Ft(2)1544 5241 y Fm(b)1525 5266 y Fs(G)p Fw(.)71 b(The)43 b(corresp)s(onding)f(inner)g(pro)s(duct)g(of)f Fs(f)5 b(;)17 b(g)47 b Ft(2)d Fq(C)3681 5213 y Ff(b)3667 5230 y Fp(G)3774 5266 y Fw(is)-360 5400 y(denoted)34 b Ft(h)16 5374 y Fw(^)56 5400 y Fs(f)5 b(;)17 b(g)t Ft(i)192 5374 y Fw(^)232 5400 y(,)30 b(the)j Fs(p)p Fw(-norm)e(is)i(denoted)g Ft(jj)1265 5374 y Fw(^)1317 5400 y Fs(f)21 b Ft(jj)1391 5374 y Fw(^)1443 5415 y Fp(p)1515 5400 y Fw(and)32 b(the)h(con)m(v)m (olution)f(of)g Fs(f)5 b(;)17 b(g)31 b Ft(2)d Fq(C)2855 5347 y Ff(b)2841 5364 y Fp(G)2939 5400 y Fw(is)k(denoted)i Fs(f)3481 5397 y Fw(^)3481 5400 y Ft(\003)16 b Fs(g)t Fw(.)p eop %%Page: 7 11 7 10 bop -180 68 a Fk(1.1.)76 b(INTR)m(ODUCTION)3128 b Fw(7)-80 400 y(Note)32 b(that)h Fs(\037)428 415 y Fp(a)469 400 y Fs(\037)530 415 y Fp(b)593 400 y Fw(=)27 b Fs(\037)757 415 y Fp(a)p Fr(+)p Fp(b)916 400 y Fw(and)33 b(hence)1774 602 y Fs(b)28 b Ft(!)f Fs(\037)2031 617 y Fp(b)3819 602 y Fw(\(1.3\))-180 827 y(is)40 b(an)h(isomorphism)d(b)s(et)m(w)m(een)43 b Fs(G)d Fw(and)1367 802 y Fm(b)1348 827 y Fs(G)p Fw(.)67 b(Hence)42 b(w)m(e)g(can)f(view)2407 801 y Fm(b)2389 827 y Fs(f)51 b Fw(as)41 b(a)f(function)g(in)g Fq(C)3283 791 y Fp(G)3389 827 y Fw(and)h(write)3862 801 y Fm(b)3844 827 y Fs(f)11 b Fw(\()p Fs(b)p Fw(\))-180 959 y(instead)33 b(of)286 932 y Fm(b)268 959 y Fs(f)10 b Fw(\()p Fs(\037)425 974 y Fp(b)460 959 y Fw(\).)-80 1079 y(F)-8 b(rom)31 b(the)i(orthogonalit)m(y)d(of)i(c)m(haracters)i(it)e(follo)m(ws)f(that) 1636 1255 y Fm(b)1618 1282 y Fs(f)11 b Fw(\()p Fs(b)p Fw(\))28 b(=)f Ft(h)p Fs(f)5 b(;)17 b(\037)2122 1297 y Fp(b)2156 1282 y Ft(i)p Fs(:)1597 b Fw(\(1.4\))-80 1484 y(Expanding)32 b(\(1.4\))g(and)h(using)f Fs(\037)1150 1499 y Fp(a)1192 1484 y Fw(\()p Fs(b)p Fw(\))c(=)g Fs(\037)1502 1499 y Fp(b)1536 1484 y Fw(\()p Fs(a)p Fw(\))33 b(w)m(e)g(obtain)1558 1705 y Fm(b)1540 1732 y Fs(f)38 b Fw(=)1744 1664 y(1)p 1739 1709 59 4 v 1739 1800 a Fs(n)1824 1637 y Fm(X)1826 1848 y Fp(a)p Fo(2)p Fp(G)1985 1732 y Fs(f)11 b Fw(\()p Fs(a)p Fw(\))p 2171 1677 103 4 v Fs(\037)2232 1747 y Fp(a)2273 1732 y Fs(:)1519 b Fw(\(1.5\))-180 2022 y Fu(Theorem)37 b(1.1.4)49 b Fj(The)34 b(F)-7 b(ourier)35 b(tr)-5 b(ansform)34 b(satis\014es)-35 2223 y Ft(\017)49 b Fj(line)-5 b(arity)469 2200 y Fq([)449 2223 y Fs(f)33 b Fw(+)22 b Fs(g)31 b Fw(=)827 2197 y Fm(b)809 2223 y Fs(f)i Fw(+)22 b Fm(b)-55 b Fs(g)s(;)1122 2197 y Fm(c)1117 2223 y Fs(af)38 b Fw(=)28 b Fs(a)1427 2197 y Fm(b)1409 2223 y Fs(f)11 b(;)51 b(f)5 b(;)17 b(g)31 b Ft(2)d Fq(C)1881 2187 y Fp(G)1947 2223 y Fs(;)51 b(a)28 b Ft(2)g Fq(C)20 b Fj(,)-35 2434 y Ft(\017)69 2408 y Fm(c)64 2434 y Fs(f)11 b(g)31 b Fw(=)322 2408 y Fm(b)304 2434 y Fs(f)380 2431 y Fw(^)379 2434 y Ft(\003)17 b Fm(b)-55 b Fs(g)s(;)582 2411 y Fq([)575 2434 y Fs(f)33 b Ft(\003)22 b Fs(g)30 b Fw(=)927 2408 y Fm(b)909 2434 y Fs(f)11 b Fm(b)-55 b Fs(g)s Fj(,)35 b Fs(f)5 b(;)17 b(g)31 b Ft(2)d Fq(C)1418 2398 y Fp(G)1483 2434 y Fj(.)-180 2623 y Fu(Pro)s(of)37 b(:)-180 2743 y Fw(The)d(linearit)m(y)c(follo)m (ws)i(from)f(the)i(linearit)m(y)e(of)h(inner)g(pro)s(duct.)44 b(T)-8 b(o)32 b(pro)m(v)m(e)2684 2717 y Fm(c)2679 2743 y Fs(f)11 b(g)31 b Fw(=)2937 2717 y Fm(b)2920 2743 y Fs(f)2995 2740 y Fw(^)2995 2743 y Ft(\003)16 b Fm(b)-55 b Fs(g)36 b Fw(note)d(that)-135 2968 y Fs(f)11 b(g)30 b Fw(=)105 2857 y Fm(\020)181 2873 y(X)183 3085 y Fp(a)p Fo(2)p Fp(G)359 2942 y Fm(b)341 2968 y Fs(f)11 b Fw(\()p Fs(a)p Fw(\))p Fs(\037)588 2983 y Fp(a)630 2857 y Fm(\021)q(\020)766 2873 y(X)771 3085 y Fp(b)p Fo(2)p Fp(G)926 2968 y Fm(b)-55 b Fs(g)s Fw(\()p Fs(b)p Fw(\))p Fs(\037)1154 2983 y Fp(b)1189 2857 y Fm(\021)1276 2968 y Fw(=)1403 2873 y Fm(X)1380 3085 y Fp(a;b)p Fo(2)p Fp(G)1586 2968 y Fs(\037)1647 2983 y Fp(a)p Fr(+)p Fp(b)1792 2942 y Fm(b)1774 2968 y Fs(f)10 b Fw(\()p Fs(a)p Fw(\))p Fm(b)-55 b Fs(g)t Fw(\()p Fs(b)p Fw(\))28 b(=)2258 2873 y Fm(X)2264 3085 y Fp(c)p Fo(2)p Fp(G)2419 2968 y Fs(\037)2480 2983 y Fp(c)2531 2873 y Fm(X)2533 3085 y Fp(a)p Fo(2)p Fp(G)2710 2942 y Fm(b)2692 2968 y Fs(f)10 b Fw(\()p Fs(a)p Fw(\))p Fm(b)-55 b Fs(g)t Fw(\()p Fs(c)22 b Ft(\000)h Fs(a)p Fw(\))k(=)3350 2873 y Fm(X)3356 3085 y Fp(c)p Fo(2)p Fp(G)3494 2968 y Fw(\()3550 2942 y Fm(b)3532 2968 y Fs(f)3608 2965 y Fw(^)3607 2968 y Ft(\003)17 b Fm(b)-56 b Fs(g)t Fw(\)\()p Fs(c)p Fw(\))p Fs(\037)3941 2983 y Fp(c)-180 3304 y Fw(and)33 b(hence)285 3278 y Fm(c)281 3304 y Fs(f)11 b(g)30 b Fw(=)539 3278 y Fm(b)521 3304 y Fs(f)597 3301 y Fw(^)596 3304 y Ft(\003)17 b Fm(b)-55 b Fs(g)s Fw(.)43 b(The)34 b(pro)s(of)e(of)1357 3281 y Fq([)1350 3304 y Fs(f)h Ft(\003)21 b Fs(g)31 b Fw(=)1702 3278 y Fm(b)1684 3304 y Fs(f)11 b Fm(b)-55 b Fs(g)36 b Fw(uses)e(\(1.5\))e(instead)g(of) g(\(1.2\).)1002 b Fi(\004)-80 3495 y Fw(An)32 b(imp)s(ortan)m(t)f(prop) s(ert)m(y)i(of)g(the)g(F)-8 b(ourier)31 b(transform)h(is)g(the)h(follo) m(wing)d(form)m(ula.)-180 3708 y Fu(Theorem)37 b(1.1.5)h(\(Planc)m (herel)d(form)m(ula\))48 b Fj(F)-7 b(or)34 b(any)h Fs(f)5 b(;)17 b(g)31 b Ft(2)d Fq(C)2327 3672 y Fp(G)1629 3922 y Ft(h)p Fs(f)5 b(;)17 b(g)t Ft(i)26 b Fw(=)i Ft(h)1992 3895 y Fw(^)8 b Fm(b)2032 3922 y Fs(f)i(;)17 b Fm(b)-55 b Fs(g)s Ft(i)2172 3895 y Fw(^)-180 4124 y Fu(Pro)s(of)37 b(:)-180 4245 y Fw(Using)32 b(the)h(orthogonalit)m(y)e(of)h(c)m (haracters)220 4459 y Ft(h)p Fs(f)5 b(;)17 b(g)t Ft(i)27 b Fw(=)g Ft(h)615 4365 y Fm(X)617 4576 y Fp(a)p Fo(2)p Fp(G)793 4433 y Fm(b)775 4459 y Fs(f)11 b Fw(\()p Fs(a)p Fw(\))p Fs(\037)1022 4474 y Fp(a)1064 4459 y Fs(;)1108 4365 y Fm(X)1114 4577 y Fp(b)p Fo(2)p Fp(G)1268 4459 y Fm(b)-55 b Fs(g)t Fw(\()p Fs(b)p Fw(\))p Fs(\037)1497 4474 y Fp(b)1531 4459 y Ft(i)28 b Fw(=)1724 4365 y Fm(X)1701 4577 y Fp(a;b)p Fo(2)p Fp(G)1925 4433 y Fm(b)1907 4459 y Fs(f)11 b Fw(\()p Fs(a)p Fw(\))p 2093 4373 168 4 v Fm(b)-55 b Fs(g)s Fw(\()p Fs(b)p Fw(\))q Ft(h)p Fs(\037)2361 4474 y Fp(a)2402 4459 y Fs(;)17 b(\037)2507 4474 y Fp(b)2541 4459 y Ft(i)28 b Fw(=)2711 4365 y Fm(X)2713 4576 y Fp(a)p Fo(2)p Fp(G)2890 4433 y Fm(b)2872 4459 y Fs(f)10 b Fw(\()p Fs(a)p Fw(\))p 3057 4373 178 4 v Fm(b)-55 b Fs(g)t Fw(\()p Fs(a)p Fw(\))28 b(=)f Ft(h)3373 4433 y Fw(^)8 b Fm(b)3413 4459 y Fs(f)i(;)17 b Fm(b)-55 b Fs(g)s Ft(i)3553 4433 y Fw(^)3593 4459 y Fs(:)3943 4758 y Fi(\004)-80 4949 y Fw(T)-8 b(aking)32 b Fs(f)38 b Fw(=)28 b Fs(g)36 b Fw(in)31 b(the)i(Planc)m(herel)g(form)m(ula)e(w)m(e)i(obtain)-180 5161 y Fu(Theorem)k(1.1.6)h(\(P)m(arsev)-6 b(al's)37 b(equalit)m(y\))47 b Fj(F)-7 b(or)34 b(any)h Fs(f)j Ft(2)28 b Fq(C)2223 5125 y Fp(G)1626 5375 y Ft(jj)p Fs(f)11 b Ft(jj)1797 5390 y Fr(2)1862 5375 y Fw(=)28 b Ft(jj)1971 5349 y Fw(^)19 b Fm(b)2023 5375 y Fs(f)i Ft(jj)2097 5349 y Fw(^)2149 5390 y Fr(2)2187 5375 y Fs(:)p eop %%Page: 8 12 8 11 bop -360 68 a Fw(8)767 b Fk(CHAPTER)34 b(1.)76 b(HARMONIC)33 b(ANAL)-8 b(YSIS)34 b(O)m(VER)f(FINITE)g(ABELIAN)h(GR)m(OUPS)-360 400 y Fw(T)-8 b(o)50 b(obtain)e(inequalities)f(for)i(other)h Fs(p)p Fw(-norms)f(w)m(e)h(will)d(need)k(the)f(Riesz-Thorin)e(in)m (terp)s(olation)f(theorem,)-360 520 y(Mink)m(o)m(wski's)34 b(and)f(H\177)-49 b(older's)32 b(inequalit)m(y)g(\(see)i([Zyg59)o(],)f (v)m(ol)f(2,)g(p.95,)h(p.94,)f(v)m(ol)g(1,)h(p.19\))-360 737 y Fu(Theorem)k(1.1.7)h(\(The)f(Riesz-Thorin)f(in)m(terp)s(olation)f (theorem\))48 b Fj(If)36 b Fs(T)51 b Fj(is)37 b(a)g(line)-5 b(ar)36 b(op)-5 b(er)g(ator)37 b(fr)-5 b(om)36 b(a)-360 857 y(me)-5 b(asur)g(e)34 b(sp)-5 b(ac)g(e)34 b Fs(A)h Fj(to)g(a)g(me)-5 b(asur)g(e)34 b(sp)-5 b(ac)g(e)35 b Fs(B)40 b Fj(such)34 b(that)1230 1070 y Ft(jj)p Fs(T)14 b(f)d Ft(jj)1472 1086 y Fr(1)p Fp(=q)1574 1095 y Fh(1)1694 1070 y Ft(\024)83 b Fs(c)1896 1085 y Fr(1)1936 1070 y Ft(jj)p Fs(f)11 b Ft(jj)2107 1086 y Fr(1)p Fp(=p)2213 1095 y Fh(1)1230 1215 y Ft(jj)p Fs(T)j(f)d Ft(jj)1472 1231 y Fr(1)p Fp(=q)1574 1240 y Fh(2)1694 1215 y Ft(\024)83 b Fs(c)1896 1230 y Fr(2)1936 1215 y Ft(jj)p Fs(f)11 b Ft(jj)2107 1231 y Fr(1)p Fp(=p)2213 1240 y Fh(2)-360 1428 y Fj(wher)-5 b(e)34 b Fw(0)28 b Ft(\024)g Fs(p)146 1443 y Fr(1)185 1428 y Fs(;)17 b(p)278 1443 y Fr(2)317 1428 y Fs(;)g(q)404 1443 y Fr(1)444 1428 y Fs(;)g(q)531 1443 y Fr(2)598 1428 y Ft(\024)28 b Fw(1)p Fj(,)34 b(then)h(for)g(any)g Fs(t)28 b Ft(2)g Fw([0)p Fs(;)17 b Fw(1])1338 1641 y Ft(jj)p Fs(T)d(f)d Ft(jj)1580 1657 y Fr(1)p Fp(=q)1715 1641 y Ft(\024)28 b Fs(c)p Ft(jj)p Fs(f)11 b Ft(jj)2033 1657 y Fr(1)p Fp(=p)-360 1854 y Fj(wher)-5 b(e)34 b Fs(p)28 b Fw(=)f Fs(tp)179 1869 y Fr(1)241 1854 y Fw(+)22 b(\(1)g Ft(\000)h Fs(t)p Fw(\))p Fs(p)670 1869 y Fr(2)709 1854 y Fj(,)35 b Fs(q)c Fw(=)d Fs(tq)1030 1869 y Fr(1)1092 1854 y Fw(+)22 b(\(1)g Ft(\000)h Fs(t)p Fw(\))p Fs(q)1515 1869 y Fr(2)1589 1854 y Fj(and)35 b Fs(c)27 b Fw(=)h Fs(c)1994 1818 y Fp(t)1994 1878 y Fr(1)2033 1854 y Fs(c)2075 1813 y Fr(1)p Fo(\000)p Fp(t)2075 1878 y Fr(2)2195 1854 y Fj(.)-360 2071 y Fu(Theorem)37 b(1.1.8)h(\(H\177)-56 b(older's)36 b(inequalit)m(y\))47 b Fj(L)-5 b(et)38 b Fs(A)f Fj(b)-5 b(e)38 b(a)f(me)-5 b(asur)g(e)37 b(sp)-5 b(ac)g(e.)52 b(L)-5 b(et)38 b Fw(1)p Fs(=p)24 b Fw(+)g(1)p Fs(=q)36 b Fw(=)d(1)p Fs(;)54 b(p;)17 b(q)36 b Ft(\025)d Fw(1)p Fj(.)-360 2191 y(F)-7 b(or)34 b(any)h Fs(f)5 b(;)17 b(g)31 b Ft(2)d Fq(C)344 2155 y Fp(A)1326 2404 y Ft(jj)p Fs(f)11 b(g)t Ft(jj)1548 2419 y Fr(1)1613 2404 y Ft(\024)28 b(jj)p Fs(f)11 b Ft(jj)1889 2419 y Fp(p)1927 2404 y Ft(jj)p Fs(g)t Ft(jj)2090 2419 y Fp(q)2126 2404 y Fs(:)-360 2617 y Fu(Theorem)37 b(1.1.9)h(\(Mink)m(o)m(wski's)e(inequalit)m(y\))47 b Fj(L)-5 b(et)50 b Fs(A)f Fj(b)-5 b(e)49 b(a)h(me)-5 b(asur)g(e)48 b(sp)-5 b(ac)g(e.)88 b(L)-5 b(et)50 b Fs(p)k Ft(\025)h Fw(1)p Fj(.)89 b(F)-7 b(or)48 b(any)-360 2737 y Fs(f)5 b(;)17 b(g)31 b Ft(2)d Fq(C)-25 2701 y Fp(A)1205 2950 y Ft(jj)p Fs(f)33 b Fw(+)22 b Fs(g)t Ft(jj)1547 2965 y Fp(p)1613 2950 y Ft(\024)28 b(jj)p Fs(f)11 b Ft(jj)1889 2965 y Fp(p)1949 2950 y Fw(+)22 b Ft(jj)p Fs(g)t Ft(jj)2210 2965 y Fp(p)2248 2950 y Fs(:)-260 3163 y Fw(An)32 b(imp)s(ortan)m(t)f (generalization)f(of)i(P)m(arsev)-5 b(al's)34 b(equalit)m(y)e(is)g(the) -360 3380 y Fu(Theorem)37 b(1.1.10)h(\(Hausdor\013-Y)-9 b(oung)38 b(inequalit)m(y\))47 b Fj(L)-5 b(et)47 b Fw(1)p Fs(=p)30 b Fw(+)h(1)p Fs(=q)52 b Fw(=)d(1)p Fs(;)63 b(p;)17 b(q)53 b Ft(\025)d Fw(1)c Fj(and)g Fs(f)60 b Ft(2)50 b Fq(C)3745 3343 y Fp(G)3810 3380 y Fj(.)-360 3500 y(Then)1066 3713 y Ft(jj)p Fs(f)11 b Ft(jj)1237 3728 y Fp(p)1303 3713 y Ft(\024)28 b(jj)1413 3687 y Fw(^)20 b Fm(b)1465 3713 y Fs(f)i Ft(jj)1540 3687 y Fw(^)1592 3728 y Fp(q)1712 3713 y Fj(for)34 b Fw(2)28 b Ft(\024)g Fs(p)g Ft(\024)g(1)1066 3872 y(jj)p Fs(f)11 b Ft(jj)1237 3887 y Fp(p)1303 3872 y Ft(\025)28 b(jj)1413 3846 y Fw(^)20 b Fm(b)1465 3872 y Fs(f)i Ft(jj)1540 3846 y Fw(^)1592 3887 y Fp(q)1737 3872 y Fj(for)35 b Fw(1)27 b Ft(\024)h Fs(p)g Ft(\024)g Fw(2)-360 4085 y Fu(Pro)s(of)37 b(:)-360 4205 y Fw(Note)c(that)697 4388 y Ft(j)743 4362 y Fm(b)725 4388 y Fs(f)11 b Fw(\()p Fs(x)p Fw(\))p Ft(j)27 b Fw(=)1074 4214 y Fm(\014)1074 4273 y(\014)1074 4333 y(\014)1074 4393 y(\014)1074 4453 y(\014)1122 4321 y Fw(1)p 1117 4365 59 4 v 1117 4456 a Fs(n)1202 4293 y Fm(X)1204 4505 y Fp(a)p Fo(2)p Fp(G)1362 4388 y Fs(f)11 b Fw(\()p Fs(a)p Fw(\))p Fs(\037)1609 4403 y Fp(x)1653 4388 y Fw(\()p Fs(a)p Fw(\))1780 4214 y Fm(\014)1780 4273 y(\014)1780 4333 y(\014)1780 4393 y(\014)1780 4453 y(\014)1841 4388 y Ft(\024)1961 4321 y Fw(1)p 1956 4365 V 1956 4456 a Fs(n)2041 4293 y Fm(X)2043 4505 y Fp(a)p Fo(2)p Fp(G)2202 4388 y Ft(j)p Fs(f)g Fw(\()p Fs(a)p Fw(\))p Ft(j)26 b Fw(=)i Ft(jj)p Fs(f)11 b Ft(jj)2745 4403 y Fr(1)-360 4676 y Fw(and)35 b(hence)h Ft(jj)110 4650 y Fw(^)20 b Fm(b)162 4676 y Fs(f)h Ft(jj)236 4650 y Fw(^)288 4691 y Fo(1)393 4676 y Ft(\024)32 b(jj)p Fs(f)11 b Ft(jj)673 4691 y Fr(1)711 4676 y Fw(.)50 b(F)-8 b(rom)33 b(P)m(arsev)-5 b(al's)36 b(equalit)m(y)e(w)m(e)i(kno)m(w)g Ft(jj)2282 4650 y Fw(^)20 b Fm(b)2334 4676 y Fs(f)h Ft(jj)2408 4650 y Fw(^)2460 4691 y Fr(2)2530 4676 y Fw(=)31 b Ft(jj)p Fs(f)11 b Ft(jj)2808 4691 y Fr(2)2846 4676 y Fw(.)50 b(An)35 b(application)d(of)j(the)-360 4797 y(Riesz-Thorin)c(in)m(terp)s (olation)e(Theorem)j(yields)f(the)h(Lemma)e(for)h(2)d Ft(\024)g Fs(p)g Ft(\024)g(1)p Fw(.)43 b(F)-8 b(or)30 b(1)e Ft(\024)g Fs(p)g Ft(\024)g Fw(2)j(note)h(that)f(for)-360 4917 y(1)p Fs(=p)22 b Fw(+)g(1)p Fs(=q)31 b Fw(=)c(1)1348 5037 y Ft(jj)p Fs(f)11 b Ft(jj)1519 4996 y Fr(2)1519 5062 y(2)1585 5037 y Fw(=)27 b Ft(jj)p Fs(f)11 b Ft(jj)1859 5052 y Fp(p)1897 5037 y Ft(jj)p Fs(f)g Ft(jj)2068 5052 y Fp(q)2105 5037 y Fs(:)3763 5209 y Fi(\004)-260 5400 y Fw(The)33 b(follo)m(wing)d(inequalities)h(for)h(norms)g(in)g Fq(C)1487 5364 y Fp(G)1584 5400 y Fw(and)h Fq(C)1854 5347 y Ff(b)1840 5364 y Fp(G)1938 5400 y Fw(will)d(b)s(e)j(useful)p eop %%Page: 9 13 9 12 bop -180 68 a Fk(1.1.)76 b(INTR)m(ODUCTION)3128 b Fw(9)-180 401 y Fu(Lemma)37 b(1.1.11)49 b Fj(L)-5 b(et)36 b Fs(p)27 b Ft(\024)h Fs(r)s Fj(.)45 b(F)-7 b(or)34 b(any)g Fs(f)39 b Ft(2)28 b Fq(C)1639 365 y Fp(G)1739 401 y Fj(and)34 b Fs(g)d Ft(2)d Fq(C)2180 348 y Ff(b)2166 365 y Fp(G)1570 623 y Ft(jj)p Fs(f)11 b Ft(jj)1741 638 y Fp(p)1862 623 y Ft(\024)84 b(jj)p Fs(f)11 b Ft(jj)2194 638 y Fp(r)2230 623 y Fs(;)1566 779 y Ft(jj)1571 753 y Fw(^)1623 779 y Fs(g)k Ft(jj)1690 753 y Fw(^)1742 794 y Fp(p)1862 779 y Ft(\025)84 b(jj)2028 753 y Fw(^)2080 779 y Fs(g)15 b Ft(jj)2147 753 y Fw(^)2199 794 y Fp(r)2235 779 y Fs(:)-180 1001 y Fu(Pro)s(of)37 b(:)-180 1122 y Fw(Note)29 b(that)g Ft(jj)p Fs(f)11 b Ft(jj)431 1137 y Fp(p)497 1122 y Fw(=)28 b Ft(jj)17 b(j)p Fs(f)11 b Ft(j)789 1085 y Fp(p)843 1122 y Ft(jj)899 1071 y Fr(1)p Fp(=p)899 1146 y Fr(1)1038 1122 y Fw(and)29 b Ft(jj)p Fs(f)11 b Ft(jj)1395 1137 y Fp(r)1459 1122 y Fw(=)28 b Ft(jj)17 b(j)p Fs(f)11 b Ft(j)1751 1085 y Fp(p)1805 1122 y Ft(jj)1861 1071 y Fr(1)p Fp(=p)1861 1153 y(r)r(=p)1971 1122 y Fw(.)42 b(Hence)31 b(w)m(e)f(can)f(w.l.o.g.)42 b(assume)30 b Fs(p)d Fw(=)h(1)h(and)g Fs(r)i(>)c Fw(1.)-180 1253 y(Let)33 b Fs(q)j Fw(b)s(e)d(the)g (conjugate)g(exp)s(onen)m(t)h(to)e Fs(r)j Fw(i.e.)43 b(1)p Fs(=r)24 b Fw(+)e(1)p Fs(=q)31 b Fw(=)d(1.)43 b(By)33 b(H\177)-49 b(older's)33 b(inequalit)m(y)1360 1474 y Ft(jj)p Fs(f)11 b Ft(jj)1531 1489 y Fp(p)1597 1474 y Ft(\024)28 b(jj)p Fs(f)11 b Ft(jj)1873 1489 y Fp(r)1909 1474 y Ft(jj)p Fu(1)p Ft(jj)2077 1489 y Fp(q)2142 1474 y Fw(=)27 b Ft(jj)p Fs(f)11 b Ft(jj)2416 1489 y Fp(r)2453 1474 y Fs(:)-180 1696 y Fw(F)-8 b(or)34 b(the)h(second)i(inequalit)m(y) c(using)i(the)g(same)g(argumen)m(t)f(w)m(e)i(can)f(assume)h Fs(p)31 b Fw(=)h(1)i(and)h Fs(r)g(>)c Fw(1.)50 b(Also)35 b(w.l.o.g.)-180 1816 y Ft(jj)p Fs(g)t Ft(jj)-17 1831 y Fr(1)48 1816 y Fw(=)28 b(1.)43 b(Hence)34 b Ft(j)p Fs(g)t Fw(\()p Fs(x)p Fw(\))p Ft(j)26 b(\024)j Fw(1)j(for)g(all)e Fs(x)e Ft(2)1492 1791 y Fm(b)1474 1816 y Fs(G)p Fw(.)43 b(Therefore)34 b Ft(j)p Fs(g)t Fw(\()p Fs(x)p Fw(\))p Ft(j)2301 1780 y Fp(r)2365 1816 y Ft(\024)28 b(j)p Fs(g)t Fw(\()p Fs(x)p Fw(\))p Ft(j)k Fw(and)h Ft(jj)2935 1790 y Fw(^)2987 1816 y Fs(g)15 b Ft(jj)3054 1790 y Fw(^)3106 1831 y Fp(r)3169 1816 y Ft(\024)28 b Fw(1.)593 b Fi(\004)-180 2040 y Fu(Theorem)37 b(1.1.12)h(\(Y)-9 b(oung's)37 b(con)m(v)m(olution) f(inequalit)m(y\))47 b Fj(L)-5 b(et)34 b Fw(1)28 b Ft(\024)g Fs(p;)17 b(q)t(;)g(r)29 b Ft(\024)g(1)p Fj(,)k Fw(1)p Fs(=r)d Fw(=)e(1)p Fs(=p)20 b Fw(+)h(1)p Fs(=q)j Ft(\000)d Fw(1)p Fj(.)-180 2160 y(F)-7 b(or)34 b(any)h Fs(f)5 b(;)17 b(g)31 b Ft(2)d Fq(C)524 2124 y Fp(G)1452 2282 y Ft(jj)p Fs(f)k Ft(\003)22 b Fs(g)t Ft(jj)1767 2297 y Fp(r)1831 2282 y Ft(\024)28 b(jj)p Fs(f)11 b Ft(jj)2107 2297 y Fp(p)2162 2282 y Ft(jj)p Fs(g)t Ft(jj)2325 2297 y Fp(q)2361 2282 y Fs(:)-180 2471 y Fu(Pro)s(of)37 b(:)-180 2591 y Fw(W)-8 b(e)40 b(ha)m(v)m(e)h Ft(jj)p Fs(f)36 b Ft(\003)27 b Fs(g)t Ft(jj)551 2606 y Fr(1)628 2591 y Ft(\024)40 b(jj)p Fs(f)11 b Ft(jj)916 2606 y Fr(1)954 2591 y Ft(jj)p Fs(g)t Ft(jj)1117 2606 y Fr(1)1194 2591 y Fw(and)39 b(from)g(H\177)-49 b(older)39 b(inequalit)m(y)f Ft(jj)p Fs(f)f Ft(\003)26 b Fs(g)t Ft(jj)2735 2606 y Fo(1)2848 2591 y Ft(\024)40 b(jj)p Fs(f)11 b Ft(jj)3136 2606 y Fo(1)3209 2591 y Ft(jj)p Fs(g)t Ft(jj)3372 2606 y Fr(1)3409 2591 y Fw(.)64 b(Hence)41 b(using)-180 2712 y(Riesz-Thorin)32 b(in)m(terp)s(olation)e(Theorem) 1459 2933 y Ft(jj)p Fs(f)i Ft(\003)22 b Fs(g)t Ft(jj)1774 2948 y Fp(p)1840 2933 y Ft(\024)28 b(jj)p Fs(f)11 b Ft(jj)2116 2948 y Fp(p)2154 2933 y Ft(jj)p Fs(g)t Ft(jj)2317 2948 y Fr(1)2354 2933 y Fs(:)1438 b Fw(\(1.6\))-180 3155 y(F)-8 b(rom)31 b(H\177)-49 b(older)32 b(inequalit)m(y)1017 3377 y Ft(jj)p Fs(f)h Ft(\003)21 b Fs(g)t Ft(jj)1332 3392 y Fo(1)1434 3377 y Ft(\024)28 b(jj)p Fs(f)11 b Ft(jj)1710 3392 y Fp(p)1748 3377 y Ft(jj)p Fs(g)t Ft(jj)1911 3392 y Fp(q)1947 3377 y Fs(;)82 b Fw(for)32 b(1)p Fs(=p)21 b Fw(+)h(1)p Fs(=q)31 b Fw(=)d(1)p Fs(:)996 b Fw(\(1.7\))-180 3598 y(No)m(w)33 b(using)f(Riesz-Thorin)g(in)m(terp)s(olation)e (Theorem)j(on)f(\(1.6\))g(and)h(\(1.7\))f(w)m(e)i(obtain)d(the)i (Lemma.)322 b Fi(\004)-80 3847 y Fw(Giv)m(en)47 b(a)h(subgroup)g Fs(H)61 b Ft(\024)55 b Fs(G)p Fw(,)c(the)d(c)m(haracters)h(for)f(whic)m (h)g Fs(\037)p Ft(j)2391 3873 y Fg(H)2506 3847 y Ft(\021)54 b Fu(1)2693 3862 y Fp(H)2808 3847 y Fw(form)47 b(a)g(subgroup)i(of)3734 3822 y Fm(b)3715 3847 y Fs(G)p Fw(.)89 b(Its)-180 3968 y(corresp)s(onding)36 b(\(via)g(\(1.3\)\))g(subgroup)h(in)e Fs(G)i Fw(is)f(denoted)h Fs(H)2148 3931 y Fo(?)2206 3968 y Fw(.)55 b(Note)37 b(that)f(the)h(c)m(haracters)h(whic)m(h)f(are)f(1)g (on)-180 4088 y Fs(H)i Fw(are)30 b(in)f(one-to-one)g(corresp)s(ondence) j(with)e(the)h(c)m(haracters)g(of)f Fs(G=H)37 b Fw(\(let)29 b Fs(\037)p Fw(\()p Fs(aH)8 b Fw(\))28 b(=)f Fs(\037)p Fw(\()p Fs(a)p Fw(\)\).)43 b(Th)m(us)32 b(w)m(e)f(ha)m(v)m(e)-180 4232 y Fs(H)-91 4196 y Fo(?)-5 4204 y Ft(\030)-4 4236 y Fw(=)113 4203 y Fq([)100 4232 y Fs(G=H)8 b Fw(.)-180 4455 y Fu(Theorem)37 b(1.1.13)h(\(P)m(oisson)e(Summation)g(F)-9 b(orm)m(ula\))48 b Fj(F)-7 b(or)34 b(a)g(sub)-5 b(gr)g(oup)35 b Fs(H)g Ft(\024)29 b Fs(G)1228 4654 y Fw(1)p 1181 4698 144 4 v 1181 4790 a Ft(j)p Fs(H)8 b Ft(j)1354 4627 y Fm(X)1351 4838 y Fp(x)p Fo(2)p Fp(H)1518 4721 y Fs(f)j Fw(\()p Fs(x)22 b Fw(+)g Fs(a)p Fw(\))28 b(=)2038 4627 y Fm(X)2010 4846 y Fp(y)r Fo(2)p Fp(H)2157 4827 y Fe(?)2245 4695 y Fm(b)2227 4721 y Fs(f)10 b Fw(\()p Fs(y)t Fw(\))p Fs(\037)2474 4736 y Fp(y)2515 4721 y Fw(\()p Fs(a)p Fw(\))p Fs(:)-180 5048 y Fu(Pro)s(of)37 b(:)223 5243 y Fw(1)p 175 5287 V 175 5378 a Ft(j)p Fs(H)8 b Ft(j)349 5216 y Fm(X)346 5427 y Fp(x)p Fo(2)p Fp(H)512 5310 y Fs(f)j Fw(\()p Fs(x)22 b Fw(+)g Fs(a)p Fw(\))28 b(=)1062 5243 y(1)p 1015 5287 V 1015 5378 a Ft(j)p Fs(H)8 b Ft(j)1188 5216 y Fm(X)1185 5427 y Fp(x)p Fo(2)p Fp(H)1352 5216 y Fm(X)1354 5427 y Fp(y)r Fo(2)p Fp(G)1530 5284 y Fm(b)1512 5310 y Fs(f)j Fw(\()p Fs(y)t Fw(\))p Fs(\037)1760 5325 y Fp(y)1800 5310 y Fw(\()p Fs(x)23 b Fw(+)f Fs(a)p Fw(\))28 b(=)2234 5216 y Fm(X)2237 5427 y Fp(y)r Fo(2)p Fp(G)2413 5284 y Fm(b)2395 5310 y Fs(f)10 b Fw(\()p Fs(y)t Fw(\))p Fs(\037)2642 5325 y Fp(y)2683 5310 y Fw(\()p Fs(a)p Fw(\))2868 5243 y(1)p 2820 5287 V 2820 5378 a Ft(j)p Fs(H)e Ft(j)2993 5216 y Fm(X)2990 5427 y Fp(x)p Fo(2)p Fp(H)3157 5310 y Fs(\037)3218 5325 y Fp(y)3259 5310 y Fw(\()p Fs(x)p Fw(\))28 b(=)g(\()p Ft(\003)p Fw(\))p Fs(:)p eop %%Page: 10 14 10 13 bop -360 68 a Fw(10)718 b Fk(CHAPTER)34 b(1.)76 b(HARMONIC)33 b(ANAL)-8 b(YSIS)34 b(O)m(VER)f(FINITE)g(ABELIAN)h(GR)m (OUPS)-360 400 y Fw(The)g(restriction)d Fs(\037)368 415 y Fp(y)410 400 y Ft(j)438 426 y Fg(H)532 400 y Fw(is)h(a)g(c)m (haracter)i(of)e Fs(H)40 b Fw(and)32 b(hence)i(b)m(y)g(the)f (orthogonalit)m(y)1082 655 y(\()p Ft(\003)p Fw(\))27 b(=)1339 561 y Fm(X)1341 772 y Fp(y)r Fo(2)p Fp(G)1517 629 y Fm(b)1499 655 y Fs(f)11 b Fw(\()p Fs(y)t Fw(\))p Fs(\037)1747 670 y Fp(y)1787 655 y Fw(\()p Fs(a)p Fw(\)\()p Fs(\037)2013 670 y Fp(y)2055 655 y Ft(j)2083 681 y Fg(H)2172 655 y Ft(\021)28 b Fu(1)p Fw(\))p Fs(:)3763 985 y Fi(\004)-260 1182 y Fw(Finally)35 b(w)m(e)k(compute)f(the)g(F)-8 b(ourier)37 b(co)s(e\016cien)m(ts)i(of)f(the)g(function)g(whic)m(h)g(has)g(v)-5 b(alue)38 b(1)f(at)h(0)g(and)g(v)-5 b(alue)37 b(0)-360 1302 y(elsewhere.)45 b(F)-8 b(or)32 b(a)g(set)h Fs(A)28 b Ft(\022)g Fs(G)33 b Fw(the)g(c)m(haracteristic)f(function)g(of)g Fs(A)h Fw(is)f(denoted)i(b)m(y)f Fu(1)2860 1317 y Fp(A)2917 1302 y Fw(,)1211 1590 y Fu(1)1267 1605 y Fp(A)1324 1590 y Fw(\()p Fs(x)p Fw(\))28 b(=)1586 1479 y Fm(n)1694 1529 y Fw(1)83 b(if)32 b Fs(x)c Ft(2)g Fs(A)1694 1649 y Fw(0)83 b(otherwise)-360 1883 y(F)-8 b(or)32 b(a)g(one)h(elemen)m(t)f(set)i Ft(f)p Fs(a)p Ft(g)e Fw(w)m(e)i(write)e Fu(1)1220 1898 y Fp(a)1294 1883 y Fw(instead)h(of)f Fu(1)1798 1899 y Fo(f)p Fp(a)p Fo(g)1910 1883 y Fw(.)-360 2134 y Fu(Example)k(1.1.14)50 b Fw(F)-8 b(or)31 b Fu(1)668 2149 y Fr(0)740 2134 y Fw(w)m(e)j(ha)m(v)m (e)1388 2287 y Fm(b)1368 2308 y Fu(1)1424 2323 y Fr(0)1491 2308 y Fw(=)1609 2241 y(1)p 1605 2286 59 4 v 1605 2377 a Fs(n)1673 2308 y(\037)1734 2323 y Fp(x)1778 2308 y Fw(\(0\))27 b(=)2048 2241 y(1)p 2044 2286 V 2044 2377 a Fs(n)-360 2526 y Fw(and)33 b(hence)1434 2681 y Fu(1)1490 2696 y Fr(0)1557 2681 y Fw(=)1675 2614 y(1)p 1671 2658 V 1671 2750 a Fs(n)1756 2587 y Fm(X)1758 2798 y Fp(a)p Fo(2)p Fp(G)1916 2681 y Fs(\037)1977 2696 y Fp(a)2019 2681 y Fs(:)-360 3102 y Fn(1.2)161 b(Represen)l(tations)51 b(of)j(Bo)t(olean)g(F)-13 b(unctions)-360 3331 y Fw(Let)23 b Fs(B)33 b Fw(=)28 b Ft(f)p Fw(0)p Fs(;)17 b Fw(1)p Ft(g)22 b Fw(where)i(0)f(represen)m(ts)j(false)d(and)g(1)g(represen)m (ts)j(true.)41 b(Giv)m(en)23 b(a)g(Bo)s(olean)f(function)h Fs(f)38 b Fw(:)28 b Fs(B)3532 3295 y Fp(n)3606 3331 y Ft(!)g Fs(B)5 b Fw(,)-360 3451 y(w)m(e)34 b(can)e(view)h(it)f(as)h(a)f (function)g Fs(f)38 b Fw(:)28 b Fq(Z)1078 3415 y Fp(n)1078 3476 y Fr(2)1150 3451 y Ft(!)g(f)p Fs(T)8 b(;)17 b(F)d Ft(g)27 b(\022)h Fq(C)20 b Fw(.)49 b(Usually)32 b(w)m(e)i(c)m(ho)s(ose) -215 3680 y Ft(\017)49 b Fs(F)41 b Fw(=)28 b(0)p Fs(;)49 b(T)41 b Fw(=)28 b(1;)k(or)-215 3909 y Ft(\017)49 b Fs(F)41 b Fw(=)28 b(1)p Fs(;)49 b(T)41 b Fw(=)28 b Ft(\000)p Fw(1.)-360 4138 y(Let)h Fs(f)-141 4153 y Fp(a)-70 4138 y Fw(b)s(e)h(the)f Ft(f)p Fw(0)p Fs(;)17 b Fw(1)p Ft(g)p Fw(-represen)m(tation)29 b(of)g Fs(f)39 b Fw(and)30 b Fs(f)1562 4153 y Fp(b)1625 4138 y Fw(b)s(e)g(the)g Ft(\006)p Fw(1)f(represen)m(tation)h(of)f Fs(f)11 b Fw(.)42 b(W)-8 b(e)29 b(ha)m(v)m(e)i Fs(f)3379 4153 y Fp(b)3441 4138 y Fw(=)d(1)15 b Ft(\000)g Fw(2)p Fs(f)3798 4153 y Fp(a)-360 4258 y Fw(and)33 b(hence)1035 4415 y Fm(b)1021 4441 y Fs(f)1069 4456 y Fp(b)1104 4441 y Fw(\()p Fs(x)p Fw(\))28 b(=)1366 4301 y Fm(\032)1482 4380 y Ft(\000)p Fw(2)1626 4354 y Fm(b)1608 4380 y Fs(f)1656 4395 y Fp(a)1698 4380 y Fw(\()p Fs(x)p Fw(\))177 b(for)32 b Fs(x)c Ft(6)p Fw(=)f(0)1482 4515 y(1)22 b Ft(\000)h Fw(2)1719 4488 y Fm(b)1702 4515 y Fs(f)1750 4530 y Fp(a)1791 4515 y Fw(\()p Fs(x)p Fw(\))84 b(for)32 b Fs(x)c Fw(=)f(0)2432 4441 y Fs(:)-360 4700 y Fw(Th)m(us)f(the)e(c)m(hoice)h(of)f(represen)m(tation)h(do)s(es)f 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m(t)i(of)e(the)h(c)m(hoice)g(of)g(the)g(represen-)-360 5400 y(tation.)p eop %%Page: 11 15 11 14 bop -180 68 a Fk(1.3.)76 b(RANDOM)32 b(RESTRICTIONS)j(AND)d(THE)h (F)m(OURIER)g(TRANSF)m(ORM)979 b Fw(11)-180 400 y Fn(1.3)161 b(Random)53 b(Restrictions)g(and)h(the)f(F)-13 b(ourier)53 b(T)-13 b(ransform)-180 619 y Fw(Random)34 b(restrictions)g(ha)m(v)m(e) j(pro)m(v)m(en)f(to)f(b)s(e)g(useful)g(in)f(the)h(analysis)f(of)h(Bo)s (olean)e(functions)i([FSS81,)g(H)-12 b(\027)-61 b(as86,)-180 739 y(LMN93].)47 b(In)35 b(this)e(section)h(w)m(e)h(will)d(pro)m(v)m(e) j(a)f(relationship)d(b)s(et)m(w)m(een)36 b(the)f(F)-8 b(ourier)32 b(co)s(e\016cien)m(ts)j(of)f(a)f(function)-180 859 y(and)23 b(its)g(random)g(restriction.)39 b(Lemma)22 b(1.3.6)h(will)e(b)s(e)j(used)g(in)f(section)g(1.4)g(to)g(analyze)h (the)g(F)-8 b(ourier)22 b(co)s(e\016cien)m(ts)-180 980 y(of)32 b(the)h Fs(AC)249 944 y Fr(0)321 980 y Fw(functions.)-180 1184 y Fu(De\014nition)j(1.3.1)49 b Fw(Let)34 b Fs(f)41 b Fw(:)31 b Fq(Z)1014 1148 y Fp(n)1014 1209 y Fr(2)1088 1184 y Ft(!)f Fq(C)60 b Fw(b)s(e)34 b(a)g(function.)47 b(Let)35 b Fs(A)30 b Ft(\022)h Fw([)p Fs(n)p Fw(])j(b)s(e)g(a)g(subset) i(of)d(v)-5 b(ariables)33 b(of)h Fs(f)11 b Fw(.)47 b(Let)34 b Fs(\013)-180 1305 y Fw(b)s(e)j(an)f(assignmen)m(t)h(of)f(v)-5 b(alues)36 b(to)g(the)h(v)-5 b(ariables)35 b(in)h Fs(A)p Fw(.)56 b(The)37 b Fu(restriction)d Fs(f)2782 1320 y Fp(A)p Fo( )p Fp(\013)2991 1305 y Fw(is)i(the)h(function)f(obtained) -180 1425 y(from)31 b Fs(f)44 b Fw(b)m(y)33 b(assigning)e(the)i(v)-5 b(alues)33 b Fs(\013)g Fw(to)f(the)h(v)-5 b(ariables)32 b(in)f Fs(A)p Fw(.)-180 1630 y Fu(De\014nition)36 b(1.3.2)49 b Fw(Let)33 b(0)27 b Ft(\024)i Fs(p)e Ft(\024)h Fw(1.)44 b(A)32 b Fs(p)p Fw(-)p Fu(random)37 b(restriction)30 b Fw(of)i Fs(f)39 b Fw(:)28 b Fq(Z)2737 1594 y Fp(n)2737 1654 y Fr(2)2809 1630 y Ft(!)f Fq(C)58 b Fw(is)33 b Fs(f)3187 1645 y Fp(A)p Fo( )p Fp(\013)3392 1630 y Fw(where)-35 1802 y Ft(\017)49 b Fs(A)29 b Fw(is)f(random)g(subset)i(of)e([)p Fs(n)p 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Fm(b)1930 2695 y Fs(f)10 b Fw(\()p Fs(x)23 b Fw(+)f Fs(y)t Fw(\))p Fs(\037)2353 2710 y Fp(\013)2401 2695 y Fw(\()p Fs(y)t Fw(\))p Fs(:)-180 2995 y Fu(Lemma)37 b(1.3.4)h(\([LMN93]\))48 b Fj(L)-5 b(et)46 b Fs(f)57 b Fw(:)47 b Fq(Z)1483 2959 y Fp(n)1483 3020 y Fr(2)1574 2995 y Ft(!)f Fq(C)20 b Fj(,)54 b Fs(A)47 b Ft(\022)g Fw([)p Fs(n)p Fw(])e Fj(and)g Fs(x)i Ft(\022)p 2697 2915 V 47 w Fs(A)p Fj(.)76 b(Then)44 b(we)h(c)-5 b(an)44 b(c)-5 b(alculate)45 b(the)-180 3133 y(fol)5 b(lowing)32 b(exp)-5 b(e)g(cte)g(d)32 b(value)g(and)h(the)g(se)-5 b(c)g(ond)32 b(moment)g(of)2013 3110 y Fq([)1997 3133 y Fs(f)2045 3148 y Fp(A)p Fo( )p Fp(\013)2251 3133 y Fj(wher)-5 b(e)32 b Fs(\013)i Fj(is)e(a)h(r)-5 b(andom)32 b Ft(f)p Fw(0)p Fs(;)17 b Fw(1)p Ft(g)p Fj(-assignment)31 b(to)-180 3254 y(variables)j(in)g Fs(A)p Fj(.)1482 3374 y Fs(E)1554 3389 y Fp(\013)1603 3374 y Fw([)1646 3351 y Fq([)1630 3374 y Fs(f)1678 3389 y Fp(A)p Fo( )p Fp(\013)1852 3374 y Fw(\()p Fs(x)p Fw(\)])28 b(=)2159 3348 y Fm(b)2141 3374 y Fs(f)11 b Fw(\()p Fs(x)p Fw(\))p Fs(;)1276 3551 y(E)1348 3566 y Fp(\013)1398 3551 y Fw([)1441 3528 y Fq([)1425 3551 y Fs(f)1473 3566 y Fp(A)p Fo( )p Fp(\013)1646 3551 y Fw(\()p Fs(x)p Fw(\))1777 3510 y Fr(2)1816 3551 y Fw(])28 b(=)1975 3457 y Fm(X)1975 3668 y Fp(y)r Fo(\022)p Fp(A)2154 3525 y Fm(b)2136 3551 y Fs(f)11 b Fw(\()p Fs(x)23 b Fw(+)f Fs(y)t Fw(\))2499 3510 y Fr(2)2537 3551 y Fs(:)-180 3827 y Fu(Pro)s(of)37 b(of)h(Lemma)f(1.3.3:)-80 4081 y Fm(X)-80 4293 y Fp(y)r Fo(\022)p Fp(A)99 4150 y Fm(b)81 4176 y Fs(f)11 b Fw(\()p Fs(x)22 b Fw(+)g Fs(y)t Fw(\))p Fs(\037)504 4191 y Fp(\013)553 4176 y Fw(\()p Fs(y)t Fw(\))27 b(=)812 4081 y Fm(X)811 4293 y Fp(y)r Fo(\022)p Fp(A)1006 4109 y Fw(1)p 983 4153 96 4 v 983 4244 a(2)1032 4215 y Fp(n)1120 4081 y Fm(X)1105 4297 y Fp(z)s Fo(\022)p Fr([)p Fp(n)p Fr(])1294 4176 y Fs(f)11 b Fw(\()p Fs(z)t Fw(\))p Fs(\037)1539 4191 y Fp(z)1580 4176 y Fw(\()p Fs(x)22 b Fw(+)g Fs(y)t Fw(\))p Fs(\037)1944 4191 y Fp(\013)1993 4176 y Fw(\()p Fs(y)t Fw(\))27 b(=)658 4421 y(1)p 561 4466 243 4 v 561 4557 a(2)610 4528 y Fp(n)p Fo(\000j)p Fp(A)p Fo(j)840 4394 y Fm(X)830 4610 y Fp(z)s Fo(2)p Fr([)p Fp(n)p Fr(])1011 4489 y Fs(f)11 b Fw(\()p Fs(z)t Fw(\))p Fs(\037)1256 4504 y Fp(z)1297 4489 y Fw(\()p Fs(x)p Fw(\))1486 4421 y(1)p 1438 4466 145 4 v 1438 4557 a(2)1487 4528 y Fo(j)p Fp(A)p Fo(j)1610 4394 y Fm(X)1609 4605 y Fp(y)r Fo(\022)p Fp(A)1771 4489 y Fs(\037)1832 4504 y Fp(y)1874 4489 y Fw(\()p Fs(z)26 b Fw(+)c Fs(\013)q Fw(\))28 b(=)2420 4421 y(1)p 2323 4466 243 4 v 2323 4557 a(2)2372 4528 y Fp(n)p Fo(\000j)p Fp(A)p Fo(j)2734 4394 y Fm(X)2592 4610 y Fp(z)s Fo(\022)p Fr([)p Fp(n)p Fr(];)p Fp(z)s Fo(\\)p Fp(A)p Fr(=)p Fp(\013)3037 4489 y Fs(f)11 b Fw(\()p Fs(z)t Fw(\))p Fs(\037)3282 4504 y Fp(z)3322 4489 y Fw(\()p Fs(x)p Fw(\))28 b(=)3601 4465 y Fq([)3585 4489 y Fs(f)3633 4504 y Fp(A)p Fo( )p Fp(\013)3806 4489 y Fw(\()p Fs(x)p Fw(\))3943 4776 y Fi(\004)-180 4967 y Fu(Pro)s(of)37 b(of)h(Lemma)f(1.3.4:)-180 5087 y Fw(Using)32 b(Lemma)f(1.3.3)209 5243 y(1)p 161 5287 145 4 v 161 5378 a(2)210 5350 y Fo(j)p Fp(A)p Fo(j)337 5216 y Fm(X)333 5427 y Fp(\013)p Fo(\022)p Fp(A)519 5287 y Fq([)502 5310 y Fs(f)550 5325 y Fp(A)p Fo( )p Fp(\013)723 5310 y Fw(\()p Fs(x)p Fw(\))d(=)1044 5243 y(1)p 996 5287 V 996 5378 a(2)1045 5350 y Fo(j)p Fp(A)p Fo(j)1172 5216 y Fm(X)1167 5427 y Fp(\013)p Fo(\022)p Fp(A)1338 5216 y Fm(X)1337 5427 y Fp(y)r Fo(\022)p Fp(A)1517 5284 y Fm(b)1499 5310 y Fs(f)10 b Fw(\()p Fs(x)23 b Fw(+)f Fs(y)t Fw(\))p Fs(\037)1922 5325 y Fp(\013)1971 5310 y Fw(\()p Fs(y)t Fw(\))k(=)2229 5216 y Fm(X)2229 5427 y Fp(y)r Fo(\022)p Fp(A)2408 5284 y Fm(b)2390 5310 y Fs(f)11 b Fw(\()p Fs(x)23 b Fw(+)f Fs(y)t Fw(\))2810 5243 y(1)p 2763 5287 V 2763 5378 a(2)2812 5350 y Fo(j)p Fp(A)p Fo(j)2938 5216 y Fm(X)2933 5427 y Fp(\013)p Fo(\022)p Fp(A)3103 5310 y Fs(\037)3164 5325 y Fp(\013)3213 5310 y Fw(\()p Fs(y)t Fw(\))27 b(=)3490 5284 y Fm(b)3472 5310 y Fs(f)10 b Fw(\()p Fs(x)p Fw(\))p Fs(:)p eop %%Page: 12 16 12 15 bop -360 68 a Fw(12)718 b Fk(CHAPTER)34 b(1.)76 b(HARMONIC)33 b(ANAL)-8 b(YSIS)34 b(O)m(VER)f(FINITE)g(ABELIAN)h(GR)m (OUPS)-360 400 y Fw(Again)d(using)i(Lemma)e(1.3.3)-219 591 y(1)p -267 635 145 4 v -267 726 a(2)-218 698 y Fo(j)p Fp(A)p Fo(j)-91 563 y Fm(X)-95 775 y Fp(\013)p Fo(\022)p Fp(A)83 635 y Fq([)74 658 y Fs(f)122 673 y Fp(a)p Fo( )p Fp(\013)280 658 y Fw(\()p Fs(x)p Fw(\))411 617 y Fr(2)478 658 y Fw(=)640 591 y(1)p 592 635 V 592 726 a(2)641 698 y Fo(j)p Fp(A)p Fo(j)768 563 y Fm(X)763 775 y Fp(\013)p Fo(\022)p Fp(A)994 563 y Fm(X)933 775 y Fp(y)968 784 y Fh(1)1002 775 y Fp(;y)1057 784 y Fh(2)1092 775 y Fo(\022)p Fp(A)1234 632 y Fm(b)1216 658 y Fs(f)11 b Fw(\()p Fs(x)22 b Fw(+)g Fs(y)1536 673 y Fr(1)1575 658 y Fw(\))1631 632 y Fm(b)1613 658 y Fs(f)11 b Fw(\()p Fs(x)22 b Fw(+)g Fs(y)1933 673 y Fr(2)1972 658 y Fw(\))p Fs(\037)2071 673 y Fp(\013)2121 658 y Fw(\()p Fs(y)2207 673 y Fr(1)2268 658 y Fw(+)g Fs(y)2414 673 y Fr(2)2453 658 y Fw(\))28 b(=)1188 868 y Fm(X)1127 1079 y Fp(y)1162 1088 y Fh(1)1197 1079 y Fp(;y)1252 1088 y Fh(2)1286 1079 y Fo(\022)p Fp(A)1428 936 y Fm(b)1410 963 y Fs(f)11 b Fw(\()p Fs(x)22 b Fw(+)g Fs(y)1730 978 y Fr(1)1769 963 y Fw(\))1825 936 y Fm(b)1807 963 y Fs(f)11 b Fw(\()p Fs(x)23 b Fw(+)f Fs(y)2128 978 y Fr(2)2167 963 y Fw(\))2263 895 y(1)p 2215 940 V 2215 1031 a(2)2264 1002 y Fo(j)p Fp(A)p Fo(j)2391 868 y Fm(X)2386 1079 y Fp(\013)p Fo(\022)p Fp(A)2556 963 y Fs(\037)2617 978 y Fp(\013)2666 963 y Fw(\()p Fs(y)2752 978 y Fr(1)2813 963 y Fw(+)g Fs(y)2959 978 y Fr(2)2998 963 y Fw(\))28 b(=)3168 868 y Fm(X)3168 1079 y Fp(y)r Fo(\022)p Fp(A)3347 936 y Fm(b)3329 963 y Fs(f)11 b Fw(\()p Fs(x)22 b Fw(+)g Fs(y)t Fw(\))3691 921 y Fr(2)3730 963 y Fs(:)3763 1252 y Fi(\004)-260 1443 y Fw(Using)28 b(the)h(Lemma)e(1.3.4)h(w)m(e)h(will) e(obtain)g(a)h(relationship)f(b)s(et)m(w)m(een)k(the)e(F)-8 b(ourier)27 b(co)s(e\016cien)m(ts)j(of)e(a)g(random)-360 1564 y(restriction)37 b(of)g Fs(f)48 b Fw(and)37 b(the)h(F)-8 b(ourier)37 b(co)s(e\016cien)m(ts)h(of)f Fs(f)11 b Fw(.)58 b(F)-8 b(or)37 b Fs(x)2027 1579 y Fp(i)2091 1564 y Ft(2)g Fs(A)h Fw(the)f(function)g Fs(f)2912 1579 y Fp(A)p Fo( )p Fp(\013)3123 1564 y Fw(is)g(not)g(a)g(function)-360 1684 y(of)j Fs(x)-186 1699 y Fp(i)-157 1684 y Fw(.)68 b(Ho)m(w)m(ev)m(er)43 b(w)m(e)e(can)g(view)g(it)f(as)h(a)f(function)h(of)f Fs(x)1798 1699 y Fp(i)1867 1684 y Fw(whic)m(h)h(do)s(es)h(not)e(dep)s (end)i(on)f Fs(x)3110 1699 y Fp(i)3138 1684 y Fw(.)68 b(Then)42 b(w)m(e)g(ha)m(v)m(e)-344 1794 y Fq([)-360 1817 y Fs(f)-312 1832 y Fp(A)p Fo( )p Fp(\013)-139 1817 y Fw(\()p Fs(y)t Fw(\))27 b(=)g(0)33 b(if)e Fs(y)36 b Fw(con)m(tains)c Fs(x)p Fw(.)-360 2026 y Fu(Lemma)37 b(1.3.5)49 b Fj(L)-5 b(et)43 b Fs(f)51 b Fw(:)41 b Fq(Z)728 1990 y Fp(n)728 2051 y Fr(2)814 2026 y Ft(!)g Fq(C)19 b Fj(.)73 b(We)42 b(c)-5 b(an)42 b(c)-5 b(ompute)41 b(the)h(exp)-5 b(e)g(cte)g(d)42 b(value)f(and)h(the)g(se)-5 b(c)g(ond)41 b(moment)g(of)-360 2147 y Fs(f)-312 2162 y Fp(A)p Fo( )p Fp(\013)-104 2147 y Fj(wher)-5 b(e)34 b Fs(A)28 b Ft( )f Fs(\013)36 b Fj(is)e(a)h Fs(p)p Fj(-r)-5 b(andom)34 b(r)-5 b(estriction.)1199 2354 y Fs(E)1271 2369 y Fp(A;\013)1394 2354 y Fw([)1437 2331 y Fq([)1421 2354 y Fs(f)1469 2369 y Fp(A)p Fo( )p Fp(\013)1642 2354 y Fw(\()p Fs(x)p Fw(\)])28 b(=)f Fs(p)1980 2313 y Fo(j)p Fp(x)p Fo(j)2081 2327 y Fm(b)2063 2354 y Fs(f)11 b Fw(\()p Fs(x)p Fw(\))p Fs(;)798 2569 y(E)870 2584 y Fp(A;\013)992 2569 y Fw([)1035 2546 y Fq([)1019 2569 y Fs(f)1067 2584 y Fp(A)p Fo( )p Fp(\013)1240 2569 y Fw(\()p Fs(x)p Fw(\))1371 2528 y Fr(2)1411 2569 y Fw(])28 b(=)f Fs(p)1618 2528 y Fo(j)p Fp(x)p Fo(j)1718 2475 y Fm(X)1724 2685 y Fp(y)r Fo(\022)p 1816 2648 40 3 v Fp(x)1896 2543 y Fm(b)1878 2569 y Fs(f)11 b Fw(\()p Fs(x)23 b Fw(+)f Fs(y)t Fw(\))2241 2528 y Fr(2)2279 2569 y Fw(\(1)g Ft(\000)h Fs(p)p Fw(\))2575 2528 y Fo(j)p Fp(y)r Fo(j)2655 2569 y Fs(:)-360 2847 y Fu(Pro)s(of)37 b(:)-360 2967 y Fw(If)e(the)g(set)g(of)f(set)h(v)-5 b(ariables)33 b Fs(A)i Fw(in)m(tersects)h Fs(x)f Fw(then)1610 2944 y Fq([)1593 2967 y Fs(f)1641 2982 y Fp(A)p Fo( )p Fp(\013)1814 2967 y Fw(\()p Fs(x)p Fw(\))d(=)e(0,)35 b(otherwise)g(b)m(y)h(Lemma)d (1.3.4,)3397 2944 y Fq([)3381 2967 y Fs(f)3429 2982 y Fp(A)p Fo( )p Fp(\013)3602 2967 y Fw(\()p Fs(x)p Fw(\))e(=)-342 3079 y Fm(b)-360 3105 y Fs(f)11 b Fw(\()p Fs(x)p Fw(\).)41 b(The)27 b(probabilit)m(y)d(that)h(no)h(elemen)m(t)f(of)g Fs(x)i Fw(is)e(c)m(hosen)i(to)e Fs(A)h Fw(is)f Fs(p)2203 3069 y Fo(j)p Fp(x)p Fo(j)2312 3105 y Fw(and)h(hence)h Fs(E)2831 3120 y Fp(A;\013)2953 3105 y Fw([)2996 3082 y Fq([)2980 3105 y Fs(f)3028 3120 y Fp(A)p Fo( )p Fp(\013)3201 3105 y Fw(\()p Fs(x)p Fw(\)])h(=)g Fs(p)3540 3069 y Fo(j)p Fp(x)p Fo(j)3641 3079 y Fm(b)3623 3105 y Fs(f)11 b Fw(\()p Fs(x)p Fw(\).)-260 3273 y(Let)31 b Fs(p)p Fw(\()p Fs(A)p Fw(\))g(denote)h(the)f(probabilit)m(y)e(that)i(the)h(set)f Fs(A)h Fw(is)e(c)m(hosen)j(in)d(the)i(random)e(restriction)g Fs(A)e Ft( )f Fs(\013)q Fw(.)43 b(Then)-277 3511 y Fs(E)-205 3526 y Fp(A;\013)-83 3511 y Fw([)-40 3488 y Fq([)-56 3511 y Fs(f)-8 3526 y Fp(A)p Fo( )p Fp(\013)165 3511 y Fw(\()p Fs(x)p Fw(\))296 3470 y Fr(2)336 3511 y Fw(])28 b(=)517 3416 y Fm(X)494 3632 y Fp(A)p Fo(\022)p Fr([)p Fp(n)p Fr(])701 3511 y Fs(p)p Fw(\()p Fs(A)p Fw(\)\()p Fs(x)22 b Ft(\\)h Fs(A)28 b Fw(=)f(0\))1411 3416 y Fm(X)1411 3628 y Fp(y)r Fo(\022)p Fp(A)1590 3484 y Fm(b)1572 3511 y Fs(f)11 b Fw(\()p Fs(x)22 b Ft([)h Fs(y)t Fw(\))1925 3470 y Fr(2)1991 3511 y Fw(=)723 3702 y Fm(X)708 3918 y Fp(y)r Fo(\022)p Fr([)p Fp(n)p Fr(])916 3770 y Fm(b)898 3796 y Fs(f)11 b Fw(\()p Fs(x)23 b Ft([)f Fs(y)t Fw(\))1251 3755 y Fr(2)1329 3702 y Fm(X)1306 3918 y Fp(A)p Fo(\022)p Fr([)p Fp(n)p Fr(])1513 3796 y Fs(p)p Fw(\()p Fs(A)p Fw(\)\()p Fs(x)g Ft(\\)h Fs(A)28 b Fw(=)f(0\)\()p Fs(y)k Ft(\022)d Fs(A)p Fw(\))f(=)2670 3702 y Fm(X)2676 3912 y Fp(y)r Fo(\022)p 2768 3875 V Fp(x)2848 3770 y Fm(b)2831 3796 y Fs(f)10 b Fw(\()p Fs(x)23 b Ft([)f Fs(y)t Fw(\))3183 3755 y Fr(2)3222 3796 y Fs(p)3271 3755 y Fo(j)p Fp(x)p Fo(j)3354 3796 y Fw(\(1)g Ft(\000)g Fs(p)p Fw(\))3649 3755 y Fo(j)p Fp(y)r Fo(j)3730 3796 y Fs(:)3763 4094 y Fi(\004)-260 4332 y Fw(Finally)38 b(w)m(e)43 b(express)h(the)d(sum)h (of)f(high)f(F)-8 b(ourier)41 b(co)s(e\016cien)m(ts)h(of)f(a)g(random)g (restriction)f(of)h Fs(f)52 b Fw(using)42 b(the)-360 4453 y(F)-8 b(ourier)47 b(co)s(e\016cien)m(ts)j(of)e Fs(f)11 b Fw(.)91 b(Let)49 b Fs(B)5 b Fw(\()p Fs(n;)17 b(p)p Fw(\))48 b(denote)h(a)f(random)g(v)-5 b(ariable)47 b(with)h(the)h(binomial)c(probabilit)m(y)-360 4573 y(distribution)31 b(with)h(parameters)g Fs(n)h Fw(\(the)g(n)m(um)m(b)s(er)g(of)f(coin)g (tosses\))i(and)f Fs(p)f Fw(\(the)h(probabilit)m(y)e(of)h(heads\).)-360 4782 y Fu(Lemma)37 b(1.3.6)49 b Fj(L)-5 b(et)35 b Fs(f)k Fw(:)28 b Fq(Z)695 4746 y Fp(n)695 4807 y Fr(2)767 4782 y Ft(!)f Fq(C)20 b Fj(.)51 b(Then)645 5094 y Fs(E)717 5109 y Fp(A;\013)856 4894 y Fm(2)856 5073 y(4)937 4999 y(X)922 5215 y Fo(j)p Fp(x)p Fo(j)p Fp(>k)1128 5071 y Fq([)1111 5094 y Fs(f)1159 5109 y Fp(A)p Fo( )p Fp(\013)1332 5094 y Fw(\()p Fs(x)p Fw(\))1463 5053 y Fr(2)1503 4894 y Fm(3)1503 5073 y(5)1597 5094 y Fw(=)1717 4999 y Fm(X)1701 5215 y Fp(x)p Fo(\022)p Fr([)p Fp(n)p Fr(])1912 5068 y Fm(b)1894 5094 y Fs(f)11 b Fw(\()p Fs(x)p Fw(\))2084 5053 y Fr(2)2123 5094 y Fs(P)2200 5013 y Fm(\000)2245 5094 y Fs(B)5 b Fw(\()p Ft(j)p Fs(x)p Ft(j)p Fs(;)17 b(p)p Fw(\))28 b Fs(>)f(k)2789 5013 y Fm(\001)-360 5400 y Fj(wher)-5 b(e)34 b Fs(A)28 b Ft( )f Fs(\013)36 b Fj(is)e(a)h Fs(p)p Fj(-r)-5 b(andom)34 b(r)-5 b(estriction.)p eop %%Page: 13 17 13 16 bop -180 79 a Fk(1.4.)76 b Fs(AC)198 43 y Fr(0)270 79 y Fk(CIR)m(CUITS)3176 b Fw(13)-180 400 y Fu(Pro)s(of)37 b(:)-180 520 y Fw(F)-8 b(or)32 b(\014xed)i Fs(A)e Fw(and)h(a)f(random)g (assignmen)m(t)g Fs(\013)h Fw(using)g(Lemma)e(1.3.4)615 833 y Fs(E)687 848 y Fp(\013)753 633 y Fm(2)753 812 y(4)834 738 y(X)819 954 y Fo(j)p Fp(x)p Fo(j)p Fp(>k)1025 810 y Fq([)1008 833 y Fs(f)1056 848 y Fp(A)p Fo( )p Fp(\013)1229 833 y Fw(\()p Fs(x)p Fw(\))1360 792 y Fr(2)1400 633 y Fm(3)1400 812 y(5)1494 833 y Fw(=)1695 738 y Fm(X)1598 965 y Fo(j)p Fp(x)p Fo(j)p Fp(>k)r(;x)p Fo(\022)p 1887 910 53 3 v Fp(A)1987 738 y Fm(X)1986 950 y Fp(y)r Fo(\022)p Fp(A)2166 807 y Fm(b)2148 833 y Fs(f)11 b Fw(\()p Fs(x)22 b Fw(+)g Fs(y)t Fw(\))2510 792 y Fr(2)2576 833 y Fw(=)2744 738 y Fm(X)2680 965 y Fo(j)p Fp(x)p Fo(\\)p 2787 910 V Fp(A)o Fo(j)p Fp(>k)2987 807 y Fm(b)2969 833 y Fs(f)10 b Fw(\()p Fs(x)p Fw(\))3158 792 y Fr(2)3198 833 y Fs(:)-180 1150 y Fw(If)33 b(w)m(e)g(let)f Fs(A)h Fw(b)s(e)f(random)g(where)i(eac) m(h)f(elemen)m(t)g(is)f(c)m(hosen)i(indep)s(enden)m(tly)f(with)g (probabilit)m(y)d Fs(p)j Fw(then)246 1463 y Fs(E)318 1478 y Fp(A;\013)457 1263 y Fm(2)457 1442 y(4)538 1368 y(X)523 1584 y Fo(j)p Fp(x)p Fo(j)p Fp(>k)729 1440 y Fq([)712 1463 y Fs(f)760 1478 y Fp(A)p Fo( )p Fp(\013)933 1463 y Fw(\()p Fs(x)p Fw(\))1064 1422 y Fr(2)1104 1263 y Fm(3)1104 1442 y(5)1198 1463 y Fw(=)1318 1368 y Fm(X)1302 1584 y Fp(x)p Fo(\022)p Fr([)p Fp(n)p Fr(])1513 1437 y Fm(b)1495 1463 y Fs(f)11 b Fw(\()p Fs(x)p Fw(\))1685 1422 y Fr(2)1724 1463 y Fs(P)j Fw(\()p Fs(x)22 b Ft(\\)p 2005 1383 74 4 v 23 w Fs(A)28 b(>)f(k)s Fw(\))h(=)2449 1368 y Fm(X)2432 1584 y Fp(x)p Fo(\022)p Fr([)p Fp(n)p Fr(])2644 1437 y Fm(b)2626 1463 y Fs(f)10 b Fw(\()p Fs(x)p Fw(\))2815 1422 y Fr(2)2855 1463 y Fs(P)2932 1382 y Fm(\000)2977 1463 y Fs(B)5 b Fw(\()p Ft(j)p Fs(x)p Ft(j)p Fs(;)17 b(p)p Fw(\))27 b Fs(>)h(k)3521 1382 y Fm(\001)3567 1463 y Fs(:)3943 1774 y Fi(\004)-180 2104 y Fn(1.4)161 b Fd(AC)404 2052 y Fw(0)510 2104 y Fn(Circuits)-180 2323 y Fw(Let)28 b(\012)g(b)s(e)g(a)g(set)g(of)f(Bo)s(olean)g(functions.)42 b(An)28 b(\012-Bo)s(olean)e(circuit)h(with)g Fs(n)h Fw(inputs)g(is)f(a) h(directed)g(acyclic)f(graph.)-180 2444 y(It)32 b(has)g(v)m(ertices)i (of)d(t)m(w)m(o)i(t)m(yp)s(es:)44 b(input)32 b(no)s(des)g Fs(x)1628 2459 y Fr(1)1668 2444 y Fs(;)17 b(:)g(:)g(:)f(;)h(x)1942 2459 y Fp(n)2021 2444 y Fw(and)32 b(gates.)44 b(One)32 b(of)f(the)i(gates)f(is)g(the)g(output)g(gate.)-180 2564 y(Eac)m(h)38 b(gate)g Fs(g)i Fw(is)d(lab)s(eled)g(b)m(y)h(a)f(function) g Fs(f)1474 2579 y Fp(g)1552 2564 y Fw(from)f(\012.)58 b(W)-8 b(e)38 b(de\014ne)h(the)f(function)f Fs(B)3041 2528 y Fp(n)3124 2564 y Ft(!)f Fs(B)42 b Fw(computed)c(b)m(y)g(a)-180 2685 y(gate)g Fs(g)k Fw(of)c(the)g(circuit)g(inductiv)m(ely)g(as)g Fs(f)1406 2700 y Fp(g)1447 2685 y Fw(\()p Fs(a)1536 2700 y Fr(1)1575 2685 y Fs(;)17 b(:)g(:)g(:)f(;)h(a)1845 2700 y Fp(k)1888 2685 y Fw(\))38 b(where)i Fs(a)2303 2700 y Fr(1)2342 2685 y Fs(;)17 b(:)g(:)g(:)f(;)h(a)2612 2700 y Fp(k)2693 2685 y Fw(are)38 b(the)h(functions)f(computed)h(b)m(y)-180 2805 y(the)28 b(predecessors)i(of)c Fs(g)t Fw(.)41 b(The)28 b(function)f(computed)h(b)m(y)g(the)f(circuit)g(is)f(the)i(function)f (computed)g(b)m(y)h(the)g(output)-180 2925 y(gate.)43 b(The)34 b(size)e Fs(M)43 b Fw(of)32 b(the)h(circuit)e(is)h(the)h(n)m (um)m(b)s(er)g(of)f(gates,)h(not)f(coun)m(ting)g(the)h(input)f(no)s (des.)44 b(The)33 b(depth)g Fs(d)-180 3046 y Fw(of)f(the)h(circuit)f (is)g(the)h(length)f(of)g(longest)g(path)h(from)e(an)h(input)h(no)s(de) f(to)h(the)g(output)f(gate.)-80 3166 y(Let)25 b Ft(f)p Fs(C)207 3181 y Fp(n)253 3166 y Ft(g)g Fw(b)s(e)g(a)f(sequence)j(of)e (circuits)f(where)i Fs(C)1704 3181 y Fp(n)1775 3166 y Fw(has)f Fs(n)g Fw(inputs.)41 b(Let)25 b Ft(f)p Fs(f)2623 3181 y Fp(n)2670 3166 y Ft(g)f Fw(b)s(e)h(the)h(corresp)s(onding)e (sequence)-180 3286 y(of)37 b(Bo)s(olean)g(functions)h(\()p Fs(f)829 3301 y Fp(n)913 3286 y Fw(is)g(the)g(function)f(computed)h(b)m (y)h Fs(C)2243 3301 y Fp(n)2289 3286 y Fw(\).)59 b(With)38 b(the)g(usual)f(abuse)i(of)e(language)g(w)m(e)-180 3407 y(sa)m(y)d(that)e(the)h(circuit)e Fs(C)40 b Fw(computes)33 b(the)g(function)f Fs(f)11 b Fw(.)-80 3527 y(An)46 b Fs(AC)243 3491 y Fr(0)328 3527 y Fw(circuit)f(consist)h(of)f(AND)h(and) g(OR)f(gates)i(of)e(un)m(b)s(ounded)i(fan-in)e(and)h(NOT)g(gates.)84 b(It)46 b(has)-180 3648 y(p)s(olynomial)33 b(size)k(and)g(constan)m(t)g (depth.)56 b(Using)37 b(deMorgan's)g(la)m(ws)g(w)m(e)h(can)e(normalize) f(the)i(circuit)e(so)i(that)-180 3768 y(it)30 b(con)m(tains)i(only)f Fs(n)g Fw(NOT)h(gates)f(connected)i(directly)e(to)g(the)h(inputs.)43 b(The)32 b(normalization)c(at)j(most)g(doubles)-180 3888 y(the)i(size)g(of)f(the)h(circuit.)-80 4009 y(It)24 b(is)g(w)m(ell)g (kno)m(wn)i(that)e(the)h Fs(AC)1111 3973 y Fr(0)1174 4009 y Fw(circuits)f(cannot)h(compute)f(parit)m(y)g([Ajt83,)g(FSS81,)h (FSS84)o(,)g(Y)-8 b(ao85)o(].)41 b(They)-180 4129 y(ev)m(en)33 b(cannot)f(appro)m(ximate)e(parit)m(y)h([Ajt83)o(].)44 b(Stronger)31 b(results)h(w)m(ere)h(sho)m(wn)f(b)m(y)h(H)-12 b(\027)-61 b(astad)31 b(and)h(Boppana)f(\(see)-180 4249 y([H)-12 b(\027)-61 b(as86],)33 b(p.)43 b(63\).)-80 4370 y(A)32 b(F)-8 b(ourier)32 b(co)s(e\016cien)m(t)h(of)f(a)g(Bo)s(olean)g (function)g Fs(f)38 b Fw(:)28 b Fq(Z)1978 4334 y Fp(n)1978 4395 y Fr(2)2050 4370 y Ft(!)f(f)p Fw(0)p Fs(;)17 b Fw(1)p Ft(g)32 b Fw(can)g(b)s(e)h(expressed)i(as)958 4556 y Fm(b)940 4583 y Fs(f)11 b Fw(\()p Fs(y)t Fw(\))26 b(=)i Fs(P)14 b Fw(\()p Fs(f)d Fw(\()p Fs(x)p Fw(\))27 b(=)h Ft(\010)1770 4598 y Fp(i)p Fo(2)p Fp(y)1882 4583 y Fs(x)1937 4598 y Fp(i)1966 4583 y Fw(\))22 b Ft(\000)h Fs(P)14 b Fw(\()p Fs(f)d Fw(\()p Fs(x)p Fw(\))27 b Ft(6)p Fw(=)g Ft(\010)2638 4598 y Fp(i)p Fo(2)p Fp(y)2751 4583 y Fs(x)2806 4598 y Fp(i)2835 4583 y Fw(\))p Fs(:)-180 4798 y Fw(Th)m(us)95 4772 y Fm(b)77 4798 y Fs(f)11 b Fw(\()p Fs(y)t Fw(\))42 b(measures)h(the)g(correlation)e(b)s(et)m(w)m(een)k Fs(f)53 b Fw(and)43 b(the)g(parit)m(y)f(of)g(v)-5 b(ariables)42 b Fs(x)3164 4813 y Fp(i)3192 4798 y Fs(;)60 b(i)45 b Ft(2)g Fs(y)t Fw(.)73 b(Since)43 b(the)-180 4918 y(functions)27 b(computed)g(b)m(y)h Fs(AC)959 4882 y Fr(0)1025 4918 y Fw(circuits)f(cannot)g(appro)m(ximate)f(the)h(parit)m(y)g(it)f(follo) m(ws)f(that)i(eac)m(h)h(high)e(F)-8 b(ourier)-180 5039 y(co)s(e\016cien)m(ts)34 b(of)e Fs(f)38 b Ft(2)28 b Fs(AC)755 5003 y Fr(0)827 5039 y Fw(m)m(ust)33 b(b)s(e)g(small.)-80 5159 y(Theorem)43 b(of)f(Linial,)h(Mansour)g(and)g(Nisan)g([LMN93])g (sho)m(ws)h(that)f(ev)m(en)h(the)g(sum)e(of)h(squares)h(of)f(high)-180 5280 y(F)-8 b(ourier)28 b(co)s(e\016cien)m(ts)j(m)m(ust)e(b)s(e)g (small.)41 b(The)30 b(rest)g(of)f(this)g(section)g(is)g(dev)m(oted)i (to)e(a)g(pro)s(of)f(of)h(their)g(result.)42 b(The)-180 5400 y(pro)s(of)32 b(is)g(a)g(sligh)m(tly)f(mo)s(di\014ed)g(v)m(ersion) j(of)e(the)h(pro)s(of)e(in)h([LMN93].)p eop %%Page: 14 18 14 17 bop -360 68 a Fw(14)718 b Fk(CHAPTER)34 b(1.)76 b(HARMONIC)33 b(ANAL)-8 b(YSIS)34 b(O)m(VER)f(FINITE)g(ABELIAN)h(GR)m (OUPS)p -194 303 3869 4 v -194 853 4 550 v -134 407 a Fu(Theorem)k(1.4.1)49 b Fj(L)-5 b(et)40 b(Bo)-5 b(ole)g(an)39 b Fs(f)49 b Fw(:)38 b Fq(Z)1398 371 y Fp(n)1398 432 y Fr(2)1480 407 y Ft(!)g(f\000)p Fw(1)p Fs(;)17 b Fw(1)p Ft(g)39 b Fj(b)-5 b(e)40 b(c)-5 b(ompute)g(d)40 b(by)h Fs(AC)2818 371 y Fr(0)2898 407 y Fj(cir)-5 b(cuit)40 b(of)g(depth)g Fs(d)-134 528 y Fj(and)35 b(size)f Fs(M)10 b Fj(.)46 b(Then)949 586 y Fm(X)941 802 y Fo(j)p Fp(x)p Fo(j)p Fp(>t)1135 654 y Fm(b)1117 680 y Fs(f)10 b Fw(\()p Fs(x)p Fw(\))1306 639 y Fr(2)1374 680 y Ft(\024)28 b Fw(2)p Fs(M)33 b Ft(\001)22 b Fw(exp)1870 540 y Fm(\022)1944 680 y Ft(\000)2054 613 y Fw(1)p 2031 657 94 4 v 2031 749 a(5)p Fs(e)2135 680 y Fw(\()p Fs(t=)p Fw(2\))2344 639 y Fr(1)p Fp(=d)2455 540 y Fm(\023)2545 680 y Fs(:)p 3671 853 4 550 v -194 856 3869 4 v -360 1059 a Fu(Remark)37 b(1)49 b Fj(The)34 b(r)-5 b(esult)35 b(pr)-5 b(ove)g(d)34 b(in)h([LMN93])g(has)f(slightly)h(di\013er)-5 b(ent)34 b(b)-5 b(ound)35 b(than)g(The)-5 b(or)g(em)33 b(1.4.1)1138 1207 y Fm(X)1131 1422 y Fo(j)p Fp(x)p Fo(j)p Fp(>t)1325 1275 y Fm(b)1307 1301 y Fs(f)10 b Fw(\()p Fs(x)p Fw(\))1496 1260 y Fr(2)1564 1301 y Ft(\024)28 b Fw(2)p Fs(M)33 b Ft(\001)21 b Fw(2)1943 1260 y Fo(\000)p Fr(\(1)p Fp(=)p Fr(20\))p Fp(t)2217 1237 y Fh(1)p Fg(=d)2322 1301 y Fs(:)-260 1642 y Fw(Theorem)26 b(1.4.1)g(has)g(in)m(teresting)g(applications)e ([LMN93],)k(for)d(example)h(functions)g(computed)g(b)m(y)h Fs(AC)3633 1606 y Fr(0)3699 1642 y Fw(cir-)-360 1763 y(cuits)g(can)g(b)s(e)g(learned)f(appro)m(ximately)g(b)m(y)h(sampling)e (their)h(v)-5 b(alue)26 b(at)h(quasip)s(olynomial)c(\(2)3068 1727 y Fr(p)r(olylog)34 b Fp(n)3372 1763 y Fw(\))26 b(randomly)-360 1883 y(c)m(hosen)g(inputs)e(\(c)m(hosen)i(under)f(the)f(uniform)e (distribution\).)39 b(Another)25 b(corollary)d([LMN93])i(of)g(Theorem)g (1.4.1)-360 2004 y(is)32 b(that)g(the)h(a)m(v)m(erage)h(sensitivit)m(y) e(of)h(a)f(function)g(computed)h(b)m(y)g(an)g Fs(AC)2377 1967 y Fr(0)2449 2004 y Fw(circuit)e(of)h(depth)i Fs(d)e Fw(is)g Fs(O)s Fw(\(\(log)16 b Fs(n)p Fw(\))3713 1967 y Fp(d)3753 2004 y Fw(\).)p -360 2081 4200 4 v -260 2204 a(Let)28 b Fs(f)40 b Fw(b)s(e)28 b(a)h(Bo)s(olean)e(function)h(on)h Fs(n)f Fw(v)-5 b(ariables,)29 b(let)e Fs(x)i Fw(b)s(e)g(an)g(assignmen) m(t)f(and)h(let)f Fs(f)11 b Fw(\()p Fs(x)p Fw(\))27 b(=)h Fs(i)p Fw(.)42 b(There)30 b(exists)-360 2325 y(smallest)37 b Fs(A)i Ft(\022)g Fw([)p Fs(n)p Fw(])h(suc)m(h)g(that)f Fs(f)893 2340 y Fp(A)p Fo( )p Fp(\013)1105 2325 y Ft(\021)g Fs(i)g Fw(where)h Fs(\013)g Fw(=)e(\()p Fs(x)27 b Ft(\\)g Fs(A)p Fw(\).)63 b(Y)-8 b(ou)39 b(can)g(view)g Fs(A)g Ft( )f Fs(\013)i Fw(as)f(a)g(pro)s(of)f(that)-360 2445 y Fs(f)11 b Fw(\()p Fs(x)p Fw(\))28 b(=)f Fs(i)p Fw(.)44 b(It)33 b(can)f(happ)s(en)i(that)e(for)g(ev)m(ery)i Fs(x)f Fw(there)h(is)e(small)e(suc)m(h)k(pro)s(of.)-360 2675 y Fu(De\014nition)i(1.4.2)49 b Fw(De\014ne)33 b(the)g Fu(non-deterministic)d Fw(decision)i(tree)i(complexit)m(y)d(of)i Fs(f)577 2900 y(D)658 2915 y Fp(i)686 2900 y Fw(\()p Fs(f)11 b Fw(\))28 b(=)73 b(max)952 2966 y Fp(x)p Fr(;)p Fp(f)7 b Fr(\()p Fp(x)p Fr(\)=)p Fp(i)1243 2900 y Fw(min)n Ft(fj)p Fs(A)p Ft(j)p Fw(;)17 b Fs(A)27 b Ft( )h Fs(\013)q(;)17 b(f)2011 2915 y Fp(A)p Fo( )p Fp(\013)2211 2900 y Ft(\021)28 b Fs(i;)17 b(x)22 b Ft(\\)h Fs(A)28 b Fw(=)f Fs(\013)q Ft(g)p Fs(:)-360 3180 y Fw(Let)33 b Fs(D)-104 3195 y Fo(\003)-65 3180 y Fw(\()p Fs(f)11 b Fw(\))27 b(=)h(max)p Ft(f)p Fs(D)514 3195 y Fr(0)553 3180 y Fw(\()p Fs(f)11 b Fw(\))p Fs(;)17 b(D)813 3195 y Fr(1)852 3180 y Fw(\()p Fs(f)11 b Fw(\))p Ft(g)p Fw(.)-260 3374 y(In)26 b([Nis91])g Fs(D)253 3389 y Fo(\003)293 3374 y Fw(\()p Fs(f)11 b Fw(\))26 b(is)g(called)f(the)i(certi\014cate)f(complexit)m(y)-8 b(.)41 b(W)-8 b(e)27 b(sa)m(y)g(that)f Fs(f)38 b Fw(is)25 b(computed)i(b)m(y)g(a)g Fs(t)p Fw(-CNF)f(\(resp.)-360 3494 y Fs(t)p Fw(-DNF\))i(form)m(ula)f(if)h Fs(f)39 b Fw(is)29 b(computed)g(b)m(y)h(a)e(CNF)h(\(resp.)44 b(DNF\))28 b(form)m(ula)f(with)h(clauses)i(of)e(size)h(at)g(most)f Fs(t)p Fw(.)43 b(Let)-360 3615 y Fs(s)-314 3630 y Fp(c)-246 3615 y Fw(\(resp.)j Fs(s)84 3630 y Fp(d)125 3615 y Fw(\))33 b(b)s(e)g(the)h(smallest)e(n)m(um)m(b)s(er)h(suc)m(h)i(that)e Fs(f)44 b Fw(can)34 b(b)s(e)f(expressed)j(as)e(an)f Fs(s)2808 3630 y Fp(c)2842 3615 y Ft(\000)p Fw(CNF)h(\(resp.)47 b Fs(s)3491 3630 y Fp(d)3531 3615 y Fw(-DNF\).)-360 3735 y(Then)34 b Fs(D)-24 3750 y Fr(0)43 3735 y Fw(=)27 b Fs(s)192 3750 y Fp(c)260 3735 y Fw(and)32 b Fs(D)530 3750 y Fr(1)597 3735 y Fw(=)c Fs(s)747 3750 y Fp(d)787 3735 y Fw(.)-260 3857 y(W)-8 b(e)32 b(sa)m(y)g(that)f(in)g(a)h(DNF)e (form)m(ula)g(clauses)i(accept)h(disjoin)m(t)d(inputs)i(if)e(for)h(eac) m(h)i(input)e(either)g(0)g(or)h(exactly)-360 3977 y(1)j(clause)g(is)f (satis\014ed.)51 b(W)-8 b(e)35 b(will)e(use)j(the)f(follo)m(wing)d (stronger)j(v)m(ersion)h(of)e(H)-12 b(\027)-61 b(astad's)36 b(Switc)m(hing)e(Lemma)g(\(see)-360 4098 y([H)-12 b(\027)-61 b(as86],)33 b(p.65\).)-360 4328 y Fu(Theorem)k(1.4.3)h(\(H)-15 b(\027)-70 b(astad's)38 b(Switc)m(hing)d(Lemma)j([H)-15 b(\027)-70 b(as86]\))49 b Fj(L)-5 b(et)25 b Fs(f)36 b Fj(b)-5 b(e)24 b(c)-5 b(ompute)g(d)25 b(by)g(a)g Fs(t)p Fj(-CNF)f(formula)-360 4448 y(and)33 b Fs(A)28 b Ft( )f Fs(\013)35 b Fj(b)-5 b(e)33 b(a)g Fs(p)p Fj(-r)-5 b(andom)33 b(r)-5 b(estriction.)44 b(With)34 b(pr)-5 b(ob)g(ability)33 b(at)h(le)-5 b(ast)34 b Fw(1)19 b Ft(\000)h Fw(\(5)p Fs(pt)p Fw(\))2724 4412 y Fp(s)2761 4448 y Fj(,)33 b Fs(f)2872 4463 y Fp(A)p Fo( )p Fp(\013)3079 4448 y Fj(is)g(c)-5 b(ompute)g(d)34 b(by)f(an)-360 4568 y Fs(s)p Fj(-DNF)h(formula)h(in)f (which)g(clauses)g(ac)-5 b(c)g(ept)35 b(disjoint)f(inputs.)-260 4809 y Fw(If)40 b(a)f(function)h Fs(g)j Fw(is)d(just)g(a)g(disjunction) f(of)h Fs(s)g Fw(atoms,)h(then)g Fm(b)-55 b Fs(g)s Fw(\()p Fs(x)p Fw(\))41 b(=)f(0)g(for)f Ft(j)p Fs(x)p Ft(j)i Fs(>)f(s)g Fw(b)s(ecause)h Fs(g)j Fw(do)s(es)c(not)-360 4929 y(dep)s(end)31 b(on)f(at)g(least)f(one)h(of)g(the)g(v)-5 b(ariables)29 b(in)g Fs(x)p Fw(.)43 b(A)30 b(function)g Fs(f)40 b Fw(whic)m(h)31 b(is)e(computed)h(b)m(y)h(an)f Fs(s)p Fw(-DNF)f(form)m(ula)-360 5050 y(in)k(whic)m(h)g(clauses)h (accept)g(disjoin)m(t)f(inputs)g(can)g(b)s(e)h(view)m(ed)g(as)g(a)f (sum)g(of)g(functions)g(whic)m(h)h(are)f(disjunctions)-360 5170 y(of)f Fs(s)h Fw(atoms)e(and)i(hence)596 5144 y Fm(b)578 5170 y Fs(f)11 b Fw(\()p Fs(x)p Fw(\))28 b(=)f(0)33 b(for)f Ft(j)p Fs(x)p Ft(j)27 b Fs(>)h(s)p Fw(.)43 b(Hence)34 b(w)m(e)f(ha)m(v)m(e)-360 5400 y Fu(Corollary)j(1.4.4)49 b Fj(L)-5 b(et)p eop %%Page: 15 19 15 18 bop -180 79 a Fk(1.4.)76 b Fs(AC)198 43 y Fr(0)270 79 y Fk(CIR)m(CUITS)3176 b Fw(15)-35 400 y Ft(\017)49 b Fs(D)145 415 y Fp(i)173 400 y Fw(\()p Fs(f)11 b Fw(\))27 b Ft(\024)h Fs(t)p Fj(,)35 b(and)-35 612 y Ft(\017)49 b Fs(A)28 b Ft( )f Fs(\013)36 b Fj(b)-5 b(e)34 b(a)h Fs(p)p Fj(-r)-5 b(andom)34 b(r)-5 b(estriction.)-180 824 y(Then)34 b Fs(D)155 839 y Fr(1)p Fo(\000)p Fp(i)273 824 y Fw(\()p Fs(f)359 839 y Fp(A)p Fo( )p Fp(\013)532 824 y Fw(\))28 b Ft(\024)g Fs(s)35 b Fj(and)f Fw(deg)18 b Fs(f)39 b Ft(\024)28 b Fs(s)35 b Fj(with)g(pr)-5 b(ob)g(ability)34 b(at)h(le)-5 b(ast)35 b Fw(1)22 b Ft(\000)g Fw(\(5)p Fs(pt)p Fw(\))2809 788 y Fp(s)2846 824 y Fj(.)-180 1018 y Fw(Using)32 b(Corollary)f(1.4.4)h(w)m(e)i(will)c(obtain)h(the)i (follo)m(wing)d(Lemma.)-180 1249 y Fu(Lemma)37 b(1.4.5)49 b Fj(L)-5 b(et)-35 1462 y Ft(\017)49 b Fs(f)c Fj(b)-5 b(e)35 b(c)-5 b(ompute)g(d)34 b(by)h Fs(AC)982 1425 y Fr(0)1057 1462 y Fj(cir)-5 b(cuit)35 b(of)f(depth)h Fs(d)g Fj(and)f(size)g Fs(M)10 b Fj(,)36 b(and)-35 1674 y Ft(\017)49 b Fs(A)28 b Ft( )f Fs(\013)36 b Fj(b)-5 b(e)34 b(a)h Fs(p)p Fj(-r)-5 b(andom)34 b(r)-5 b(estriction.)-180 1886 y(Then)34 b(for)h(any)g Fs(s)27 b Ft(\025)h Fw(1)35 b Fj(with)g(pr)-5 b(ob)g(ability)34 b(at)h(le)-5 b(ast)35 b Fw(1)22 b Ft(\000)g Fs(M)10 b Fw(\(5)p Fs(p)2109 1850 y Fr(1)p Fp(=d)2221 1886 y Fs(s)2267 1850 y Fr(1)p Fo(\000)p Fr(1)p Fp(=d)2468 1886 y Fw(\))2506 1850 y Fp(s)1595 2112 y Fw(deg)q(\()p Fs(f)1828 2127 y Fp(A)p Fo( )p Fp(\013)2001 2112 y Fw(\))28 b Ft(\024)g Fs(s:)-180 2339 y Fu(Pro)s(of)37 b(:)-180 2459 y Fw(Let)c Fs(f)43 2474 y Fr(1)82 2459 y Fs(;)17 b(:)g(:)g(:)f(;)h(f)349 2474 y Fp(k)424 2459 y Fw(b)s(e)33 b(functions)f(suc)m(h)i(that)f Fs(D)1490 2474 y Fo(\003)1529 2459 y Fw(\()p Fs(f)1615 2474 y Fp(i)1643 2459 y Fw(\))28 b Ft(\024)g Fs(t)g Ft(\024)g Fs(s;)49 b(i)28 b Ft(2)g Fw([)p Fs(k)s Fw(].)44 b(Observ)m(e)35 b(that)1059 2712 y Fs(D)1140 2727 y Fr(0)1180 2602 y Fm(\020)1275 2618 y(^)1256 2833 y Fp(i)p Fo(2)p Fr([)p Fp(k)r Fr(])1421 2712 y Fs(f)1469 2727 y Fp(i)1497 2602 y Fm(\021)1585 2712 y Ft(\024)28 b Fs(t)390 b(D)2196 2727 y Fr(1)2236 2602 y Fm(\020)2331 2618 y(_)2312 2833 y Fp(i)p Fo(2)p Fr([)p Fp(k)r Fr(])2477 2712 y Fs(f)2525 2727 y Fp(i)2553 2602 y Fm(\021)2640 2712 y Ft(\024)29 b Fs(t)-180 3048 y Fw(b)s(ecause)i(to)f(pro)m(v)m(e)h(that)f Ft(^)p Fs(f)878 3063 y Fp(i)936 3048 y Fw(is)g(zero)g(\(or)g(that)f Ft(_)p Fs(f)1712 3063 y Fp(i)1771 3048 y Fw(is)g(one\))h(it)f(is)h (enough)g(to)g(pro)m(v)m(e)h(that)f(an)m(y)g(of)g(the)g Fs(f)3693 3063 y Fp(i)3751 3048 y Fw(is)g(zero)-180 3168 y(\(or)i(that)h(an)m(y)g(of)f(the)h Fs(f)700 3183 y Fp(i)761 3168 y Fw(is)f(one\).)43 b(If)33 b Fs(A)28 b Ft( )f Fs(\013)33 b Fw(is)f(a)h Fs(q)t Fw(-random)e(restriction)g(then)i(b)m(y)h(H)-12 b(\027)-61 b(astad's)33 b(Lemma)757 3421 y Fs(D)838 3436 y Fo(\003)877 3311 y Fm(\020\020)1032 3327 y(^)1013 3542 y Fp(i)p Fo(2)p Fr([)p Fp(k)r Fr(])1178 3421 y Fs(f)1226 3436 y Fp(i)1254 3311 y Fm(\021)1314 3490 y Fp(A)p Fo( )p Fp(\013)1487 3311 y Fm(\021)1574 3421 y Ft(\024)28 b Fs(s)390 b(D)2196 3436 y Fo(\003)2236 3311 y Fm(\020\020)2390 3327 y(_)2371 3542 y Fp(i)p Fo(2)p Fr([)p Fp(k)r Fr(])2537 3421 y Fs(f)2585 3436 y Fp(i)2613 3311 y Fm(\021)2672 3490 y Fp(A)p Fo( )p Fp(\013)2845 3311 y Fm(\021)2932 3421 y Ft(\024)29 b Fs(s)735 b Fw(\(1.8\))-180 3757 y(where)34 b(eac)m(h)f(of)f(the)h(t)m(w)m(o)h(ev)m(en)m(ts)h(happ)s(ens)e(with)f (probabilit)m(y)f(at)h(least)g(1)22 b Ft(\000)h Fw(\(5)p Fs(q)t(t)p Fw(\))2909 3721 y Fp(s)2945 3757 y Fw(.)-80 3879 y(Let)30 b Fs(V)149 3894 y Fp(i)208 3879 y Fw(denote)h(the)g(set)g (of)f(no)s(des)h(of)f(heigh)m(t)g Fs(i)g Fw(in)g(the)h(circuit.)41 b(The)32 b(functions)e(computed)h(b)m(y)g(the)g(lea)m(v)m(es)g(of)-180 4000 y(the)k(circuit)e(ha)m(v)m(e)j Fs(D)603 4015 y Fo(\003)673 4000 y Fw(=)30 b(1.)49 b(By)34 b(\(1.8\))g(after)g(applying)f Fs(q)t(s)p Fw(-random)g(restriction,)h(all)e(functions)i(computed)h(b)m (y)-180 4120 y(the)e(no)s(des)g(of)f(heigh)m(t)h(1)f(will)e(ha)m(v)m(e) k Fs(D)1240 4135 y Fo(\003)1307 4120 y Ft(\024)28 b Fs(s)33 b Fw(with)f(probabilit)m(y)f(at)h(least)g(1)22 b Ft(\000)h(j)p Fs(V)2815 4135 y Fr(1)2854 4120 y Ft(j)p Fw(\(5)p Fs(q)t(s)p Fw(\))3100 4084 y Fp(s)3135 4120 y Fw(.)-80 4242 y(No)m(w)44 b(inductiv)m(ely)-8 b(,)46 b(if)c(the)i(functions)f(computed)h(b)m(y)g (the)g(no)s(des)g(of)f(heigh)m(t)h Fs(i)f Fw(ha)m(v)m(e)i Fs(D)3290 4257 y Fo(\003)3376 4242 y Ft(\024)i Fs(s)c Fw(then)h(after)-180 4362 y(applying)32 b(a)g Fs(q)t Fw(-random)g(restriction,)g(the)i(functions)f(computed)g(b)m(y)h(the)f (no)s(des)h(of)e(heigh)m(t)h Fs(i)23 b Fw(+)f(1)33 b(ha)m(v)m(e)h Fs(D)3800 4377 y Fo(\003)3868 4362 y Ft(\024)29 b Fs(s)-180 4483 y Fw(with)j(probabilit)m(y)f(at)h(least)g(1)22 b Ft(\000)h(j)p Fs(V)1144 4498 y Fp(i)p Fr(+1)1262 4483 y Ft(j)p Fw(\(5)p Fs(q)t(s)p Fw(\))1508 4447 y Fp(s)1543 4483 y Fw(.)-80 4605 y(After)37 b(applying)f(a)h Fs(q)t Fw(-random)e(restriction)h Fs(d)25 b Ft(\000)h Fw(2)37 b(times)f(w)m(e)i(obtain)e(that)h(with)g(probabilit)m(y)e(at)i(least)f (1)26 b Ft(\000)-180 4725 y Fw(\()p Fs(M)35 b Ft(\000)25 b Fw(1\)\(5)p Fs(q)t(s)p Fw(\))394 4689 y Fp(s)466 4725 y Fw(the)36 b(functions)g(computed)g(b)m(y)h(the)f(no)s(des)h(of)e (heigh)m(t)h Fs(d)24 b Ft(\000)h Fw(1)36 b(ha)m(v)m(e)h Fs(D)3086 4740 y Fo(\003)3159 4725 y Ft(\024)d Fs(s)p Fw(.)53 b(After)36 b(applying)-180 4846 y(one)j(more)f Fs(q)t Fw(-random)f(restriction)h(w)m(e)i(obtain)e(that)g(the)h (function)g(computed)g(b)m(y)g(the)h(circuit)d(has)i(deg)h Ft(\024)f Fs(s)-180 4966 y Fw(with)e(probabilit)m(y)f(at)h(least)h(1)25 b Ft(\000)h Fs(M)10 b Fw(\(5)p Fs(q)t(s)p Fw(\))1408 4930 y Fp(s)1445 4966 y Fw(.)59 b(Hence)39 b(w)m(e)g(obtained)e(that)g (for)h(a)f(function)g Fs(f)48 b Fw(computed)38 b(b)m(y)h(an)-180 5087 y Fs(AC)-30 5050 y Fr(0)10 5087 y Fw([)p Fs(d)p Fw(])30 b(circuit)f(of)h(size)h Fs(M)41 b Fw(and)31 b Fs(sq)1154 5050 y Fp(d)1194 5087 y Fw(-random)f(restriction)f Fs(A)f Ft( )f Fs(\013)q Fw(,)k(deg)18 b Fs(f)2608 5102 y Fp(A)p Fo( )p Fp(\013)2809 5087 y Ft(\024)28 b Fs(s)i Fw(with)h(probabilit)m(y)d(at)i(least)-180 5207 y(1)22 b Ft(\000)h Fs(M)10 b Fw(\(5)p Fs(q)t(s)p Fw(\))313 5171 y Fp(s)350 5207 y Fw(.)3566 b Fi(\004)-80 5400 y Fw(W)-8 b(e)33 b(will)d(use)j(Cherno\013)7 b('s)34 b(b)s(ound)f(to)f(estimate)g Fs(B)5 b Fw(\()p Fs(n;)17 b(p)p Fw(\))32 b(\(see)i([MR95],)f(p.)43 b(235\):)p eop %%Page: 16 20 16 19 bop -360 68 a Fw(16)718 b Fk(CHAPTER)34 b(1.)76 b(HARMONIC)33 b(ANAL)-8 b(YSIS)34 b(O)m(VER)f(FINITE)g(ABELIAN)h(GR)m (OUPS)-360 400 y Fu(Theorem)j(1.4.6)h(\(Cherno\013)10 b('s)38 b(Bound\))49 b Fj(L)-5 b(et)29 b Fs(X)1631 415 y Fr(1)1671 400 y Fs(;)17 b(:)g(:)g(:)e(;)i(X)1970 415 y Fp(n)2046 400 y Fj(b)-5 b(e)28 b(indep)-5 b(endent)28 b(r)-5 b(andom)28 b(variables.)41 b(L)-5 b(et)30 b Fs(X)35 b Fw(=)-360 520 y Fs(X)-279 535 y Fr(1)-217 520 y Fw(+)22 b Ft(\001)17 b(\001)g(\001)j Fw(+)i Fs(X)198 535 y Fp(n)280 520 y Fj(and)34 b Fs(\026)27 b Fw(=)h Fs(E)6 b Fw([)p Fs(X)i Fw(])p Fj(.)45 b(F)-7 b(or)34 b Fw(0)27 b Ft(\024)h Fs(\016)k Ft(\024)c Fw(1)1147 763 y Fs(P)1224 682 y Fm(\000)1269 763 y Fs(X)35 b Ft(\024)28 b Fw(\(1)22 b Ft(\000)h Fs(\016)t Fw(\))p Fs(\026)1843 682 y Fm(\001)1916 763 y Ft(\024)28 b Fs(e)2066 722 y Fo(\000)p Fp(\026\016)2196 699 y Fh(2)2232 722 y Fp(=)p Fr(2)2306 763 y Fs(:)-360 1001 y Fu(Pro)s(of)37 b(of)h(Theorem)f(1.4.1)-360 1122 y Fw(Let)c Fs(p)27 b Ft(2)h Fw([0)p Fs(;)17 b Fw(1])32 b(and)h Fs(s)28 b Ft(\025)g Fw(1.)43 b(By)33 b(Lemma)e(1.3.6)629 1378 y Fm(X)617 1593 y Fp(x)p Fo(2)p Fr([)p Fp(n)p Fr(])820 1446 y Fm(b)802 1472 y Fs(f)11 b Fw(\()p Fs(x)p Fw(\))992 1431 y Fr(2)1032 1472 y Fs(P)1109 1391 y Fm(\000)1154 1472 y Fs(B)5 b Fw(\()p Ft(j)p Fs(x)p Ft(j)p Fs(;)17 b(p)p Fw(\))27 b Fs(>)h(s)1690 1391 y Fm(\001)1763 1472 y Fw(=)f Fs(E)1938 1487 y Fp(A;\013)2077 1272 y Fm(2)2077 1451 y(4)2172 1378 y(X)2144 1595 y Fo(j)p Fp(x)p Fo(j)p Fp(>s)2312 1576 y Fh(2)2379 1446 y Fm(b)2361 1472 y Fs(f)2409 1487 y Fp(A)p Fo( )p Fp(\013)2582 1472 y Fw(\()p Fs(x)p Fw(\))2713 1431 y Fr(2)2753 1272 y Fm(3)2753 1451 y(5)2836 1472 y Fs(:)-360 1823 y Fw(F)-8 b(or)31 b(a)g(Bo)s(olean)g Fs(f)11 b Fw(,)32 b Ft(jj)p Fs(f)11 b Ft(jj)558 1838 y Fr(2)623 1823 y Fw(=)28 b(1)j(and)h(hence)h(b)m(y)g(the)f(Planc)m (herel)g(form)m(ula)e(the)i(random)f(v)-5 b(ariable)30 b(in)h(the)h(paren-)-360 1943 y(thesis)f(has)g(v)-5 b(alue)30 b(at)h(most)f(1.)43 b(Moreo)m(v)m(er)32 b(b)m(y)f(Lemma)f(1.4.5)g(with) g(probabilit)m(y)f(at)h(least)h(1)18 b Ft(\000)h Fs(M)10 b Fw(\(5)p Fs(p)3407 1907 y Fr(1)p Fp(=d)3518 1943 y Fs(s)3564 1907 y Fr(1)p Fo(\000)p Fr(1)p Fp(=d)3765 1943 y Fw(\))3803 1907 y Fp(s)-360 2064 y Fw(it)32 b(has)h(v)-5 b(alue)32 b(0.)43 b(Th)m(us)777 2103 y Fm(X)764 2319 y Fp(x)p Fo(2)p Fr([)p Fp(n)p Fr(])968 2171 y Fm(b)950 2198 y Fs(f)10 b Fw(\()p Fs(x)p Fw(\))1139 2156 y Fr(2)1179 2198 y Fs(P)1256 2117 y Fm(\000)1301 2198 y Fs(B)5 b Fw(\()p Ft(j)p Fs(x)p Ft(j)p Fs(;)17 b(p)p Fw(\))27 b Fs(>)h(s)1837 2117 y Fm(\001)1910 2198 y Ft(\024)g Fs(M)10 b Fw(\(5)p Fs(p)2255 2156 y Fr(1)p Fp(=d)2367 2198 y Fs(s)2413 2156 y Fr(1)p Fo(\000)p Fr(1)p Fp(=d)2614 2198 y Fw(\))2652 2156 y Fp(s)2689 2198 y Fs(:)-360 2498 y Fw(Since)33 b Fs(P)14 b Fw(\()p Fs(B)5 b Fw(\()p Fs(t;)17 b(p)p Fw(\))27 b Fs(>)g(s)p Fw(\))33 b(is)f(an)g(increasing)g(function) g(of)g Fs(t)h Fw(w)m(e)g(obtain)1118 2703 y Fm(X)1111 2919 y Fo(j)p Fp(x)p Fo(j)p Fp(>t)1305 2772 y Fm(b)1287 2798 y Fs(f)10 b Fw(\()p Fs(x)p Fw(\))1476 2757 y Fr(2)1544 2798 y Ft(\024)1659 2730 y Fs(M)g Fw(\(5)p Fs(p)1899 2694 y Fr(1)p Fp(=d)2010 2730 y Fs(s)2056 2694 y Fr(1)p Fo(\000)p Fr(1)p Fp(=d)2258 2730 y Fw(\))2296 2694 y Fp(s)p 1659 2775 674 4 v 1682 2872 a Fs(P)1759 2791 y Fm(\000)1804 2872 y Fs(B)5 b Fw(\()p Fs(t;)17 b(p)p Fw(\))27 b Fs(>)h(s)2264 2791 y Fm(\001)2342 2798 y Fs(:)-360 3142 y Fw(Let)33 b Fs(p)27 b Fw(=)h(2)p Fs(s=t)k Fw(and)h Fs(s)27 b Ft(\025)i Fw(3.)43 b(By)33 b(Cherno\013)7 b('s)34 b(Bound)960 3417 y Fs(P)1037 3306 y Fm(\020)1096 3417 y Fs(B)5 b Fw(\()p Fs(t;)17 b(p)p Fw(\))28 b Fs(>)1520 3350 y(tp)p 1520 3394 85 4 v 1538 3485 a Fw(2)1615 3306 y Fm(\021)1702 3417 y Ft(\025)g Fw(1)22 b Ft(\000)g Fs(e)2022 3376 y Fo(\000)p Fp(tp=)p Fr(8)2241 3417 y Ft(\025)28 b Fw(1)p Fs(=)p Fw(2)p Fs(:)-360 3693 y Fw(F)-8 b(or)32 b Fs(s)27 b Fw(=)18 3654 y Fr(1)p 2 3670 69 4 v 2 3727 a(5)p Fp(e)80 3693 y Fw(\()p Fs(t=)p Fw(2\))289 3657 y Fr(1)p Fp(=d)432 3693 y Fw(w)m(e)34 b(obtain)932 3898 y Fm(X)925 4114 y Fo(j)p Fp(x)p Fo(j)p Fp(>t)1118 3967 y Fm(b)1101 3993 y Fs(f)10 b Fw(\()p Fs(x)p Fw(\))1290 3952 y Fr(2)1358 3993 y Ft(\024)28 b Fw(2)p Fs(M)k Ft(\001)22 b Fw(exp)1854 3853 y Fm(\022)1927 3993 y Ft(\000)2037 3926 y Fw(1)p 2014 3970 94 4 v 2014 4061 a(5)p Fs(e)2119 3993 y Fw(\()p Fs(t=)p Fw(2\))2328 3952 y Fr(1)p Fp(=d)2438 3853 y Fm(\023)2528 3993 y Fs(:)3763 4330 y Fi(\004)-360 4690 y Fn(1.5)161 b(The)53 b(In\015uence)f(of)h(V)-13 b(ariables)-360 4918 y Fw(The)49 b Fu(in\015uence)f Fw(of)g(a)g(v)-5 b(ariable)46 b Fs(x)996 4933 y Fp(i)1073 4918 y Fw(on)i(a)f(Bo)s(olean) g(function)h Fs(f)58 b Fw(is)48 b(the)h(probabilit)m(y)d(that)i(for)f (a)h(random)-360 5039 y(assignmen)m(t)29 b Fs(A)f Ft( )f Fs(\013)q(;)46 b(A)28 b Fw(=)f([)p Fs(n)p Fw(])16 b Ft(n)g(f)p Fs(i)p Ft(g)29 b Fw(of)g(v)-5 b(alues)30 b(to)f(the)h(other)f(v)-5 b(ariables)29 b(the)g(function)g Fs(f)2984 5054 y Fp(A)p Fo( )p Fp(\013)3187 5039 y Fw(is)g(not)g(constan)m(t.)-360 5159 y(F)-8 b(or)39 b(the)i(AND)f(function)g(ev)m(ery)i(v)-5 b(ariable)38 b(has)j(a)f(tin)m(y)g(in\015uence)h(2)2229 5123 y Fr(1)p Fo(\000)p Fp(n)2366 5159 y Fw(.)67 b(Ho)m(w)m(ev)m(er)42 b(if)d(the)i(function)f(is)g(more)-360 5280 y(balanced)i(e.g.)70 b(if)41 b Fs(f)52 b Fw(is)41 b(zero)h(on)g(a)f(half)g(of)g(the)h (inputs)g(then)g(there)g(is)g(a)f(v)-5 b(ariable)40 b(with)h(a)h(large) e(in\015uence)-360 5400 y(\012\(log)17 b Fs(n=n)p Fw(\).)44 b(This)32 b(result)h(is)f(due)h(to)f(Kahn,)h(Kalai)d(and)j(Linial)d ([KKL88)o(].)p eop %%Page: 17 21 17 20 bop -180 68 a Fk(1.5.)76 b(THE)33 b(INFLUENCE)h(OF)e(V)-11 b(ARIABLES)2280 b Fw(17)p -14 303 3869 4 v -14 1572 4 1269 v 46 403 a Fu(Theorem)38 b(1.5.1)49 b Fj(F)-7 b(or)34 b(a)g(function)h Fs(f)j Fw(:)28 b Fq(Z)1661 367 y Fp(n)1661 428 y Fr(2)1733 403 y Ft(!)f(f)p Fw(0)p Fs(;)17 b Fw(1)p Ft(g)34 b Fj(such)h(that)g Fs(P)14 b Fw(\()p Fs(f)38 b Fw(=)27 b(1\))h(=)f Fs(p)h Ft(\024)g Fw(1)p Fs(=)p Fw(2)1414 587 y Fm(X)1409 803 y Fp(i)p Fo(2)p Fr([)p Fp(n)p Fr(])1579 682 y Fs(I)1622 697 y Fp(i)1650 682 y Fw(\()p Fs(f)11 b Fw(\))1785 641 y Fr(2)1852 682 y Ft(\025)28 b Fs(c)1999 697 y Fr(1)2039 682 y Fs(p)2088 641 y Fr(2)2137 615 y Fw(\(log)17 b Fs(n)p Fw(\))2414 579 y Fr(2)p 2137 659 317 4 v 2266 750 a Fs(n)46 1013 y Fj(in)35 b(p)-5 b(articular)35 b(ther)-5 b(e)35 b(is)f Fs(i)28 b Ft(2)g Fw([)p Fs(n)p Fw(])36 b Fj(such)e(that)1591 1282 y Fs(I)1634 1297 y Fp(i)1663 1282 y Fw(\()p Fs(f)11 b Fw(\))27 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Fp(")1512 2259 y Fw(\()p Fs(f)11 b Fw(\))27 b(=)1794 2165 y Fm(X)1778 2380 y Fp(x)p Fo(\022)p Fr([)p Fp(n)p Fr(])1989 2233 y Fm(b)1971 2259 y Fs(f)11 b Fw(\()p Fs(x)p Fw(\))p Fs(")2207 2218 y Fo(j)p Fp(x)p Fo(j)2290 2259 y Fs(\037)2351 2274 y Fp(x)2395 2259 y Fs(:)1397 b Fw(\(1.9\))-180 2594 y(In)33 b([Bec75])g(it)f(w)m(as)h(sho)m(wn)h(that)e Fs(T)1131 2609 y Fp(")1201 2594 y Fw(is)g(a)g(norm)g(1)g(op)s(erator)g (from)f Fs(L)2405 2558 y Fr(1+)p Fp(")2528 2535 y Fh(2)2567 2594 y Fw(\()p Fq(Z)2674 2558 y Fp(n)2674 2619 y Fr(2)2719 2594 y Fw(\))h(to)g Fs(L)2974 2558 y Fr(2)3014 2594 y Fw(\()p Fq(Z)3121 2558 y Fp(n)3121 2619 y Fr(2)3165 2594 y Fw(\).)-180 2809 y Fu(Lemma)37 b(1.5.2)49 b Fj(F)-7 b(or)34 b(any)h Fs(f)j Fw(:)28 b Fq(Z)1075 2773 y Fp(n)1075 2833 y Fr(2)1147 2809 y Ft(!)g Fq(C)1523 3016 y Ft(jj)p Fs(T)1636 3031 y Fp(")1672 3016 y Fs(f)11 b Ft(jj)1787 3031 y Fr(2)1854 3016 y Ft(\024)28 b(jj)p Fs(f)11 b Ft(jj)2130 3033 y Fr(1+)p Fp(")2253 3014 y Fh(2)2290 3016 y Fs(:)-80 3224 y Fw(W)-8 b(e)33 b(will)d(pro)m(v)m(e)k(Lemma)d(1.5.2)h(using)g (Bec)m(kner's)j(Lemmas)d(in)g(section)g(1.6.)-180 3415 y Fu(Pro)s(of)37 b(of)h(Theorem)f(1.5.1)-180 3536 y Fw(Let)c Fs(f)43 3551 y Fp(i)71 3536 y Fw(\()p Fs(x)p Fw(\))28 b(=)g Fs(f)11 b Fw(\()p Fs(x)p Fw(\))22 b Ft(\000)g Fs(f)11 b Fw(\()p Fs(x)23 b Fw(+)f Fs(i)p Fw(\).)43 b(Clearly)32 b(for)g(an)m(y)h Fs(p)p Fw(,)g Fs(I)1883 3551 y Fp(i)1911 3536 y Fw(\()p Fs(f)11 b Fw(\))27 b(=)h Ft(jj)p Fs(f)2281 3551 y Fp(i)2308 3536 y Ft(jj)2364 3499 y Fp(p)2364 3560 y(p)2403 3536 y Fw(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)76 3795 y Fm(b)66 3821 y Fs(f)114 3836 y Fp(i)142 3821 y Fw(\()p Fs(x)p Fw(\))28 b(=)438 3753 y(1)p 415 3798 96 4 v 415 3889 a(2)464 3860 y Fp(n)552 3726 y Fm(X)537 3942 y Fp(y)r Fo(\022)p Fr([)p Fp(n)p Fr(])728 3740 y Fm(\000)773 3821 y Fs(f)11 b Fw(\()p Fs(y)t Fw(\))21 b Ft(\000)i Fs(f)11 b Fw(\()p Fs(y)25 b Fw(+)d Fs(i)p Fw(\))1420 3740 y Fm(\001)1466 3821 y Fw(\()p Ft(\000)p Fw(1\))1668 3780 y Fo(j)p Fp(x)p Fo(\\)p 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Fw(=)28 b Ft(jj)p Fs(T)1674 4743 y Fp(")1710 4728 y Fs(f)1758 4743 y Fp(i)1786 4728 y Ft(jj)1842 4686 y Fr(2)1842 4752 y(2)1909 4728 y Ft(\024)g(jj)p Fs(f)2118 4743 y Fp(i)2145 4728 y Ft(jj)2201 4686 y Fr(2)2201 4753 y(1+)p Fp(")2324 4734 y Fh(2)2390 4728 y Fw(=)f Fs(I)2536 4743 y Fp(i)2565 4728 y Fw(\()p Fs(f)11 b Fw(\))2700 4686 y Fr(2)p Fp(=)p Fr(\(1+)p Fp(")2920 4663 y Fh(2)2955 4686 y Fr(\))2986 4728 y Fs(:)758 b Fw(\(1.10\))-180 5027 y(Summing)31 b(\(1.10\))g(o)m(v)m(er)j Fs(i)28 b Ft(2)g Fw([)p Fs(n)p Fw(])33 b(w)m(e)h(obtain)1127 5252 y(4)1209 5158 y Fm(X)1193 5373 y Fp(x)p Fo(\022)p Fr([)p Fp(n)p Fr(])1386 5252 y Ft(j)p Fs(x)p Ft(j)1515 5226 y Fm(b)1497 5252 y Fs(f)10 b Fw(\()p Fs(x)p Fw(\))1686 5211 y Fr(2)1726 5252 y Fs(")1772 5211 y Fr(2)p Fo(j)p Fp(x)p Fo(j)1918 5252 y Ft(\024)2028 5158 y Fm(X)2023 5373 y Fp(i)p Fo(2)p Fr([)p Fp(n)p Fr(])2193 5252 y Fs(I)2236 5267 y Fp(i)2264 5252 y Fw(\()p Fs(f)h Fw(\))2399 5211 y Fr(2)p Fp(=)p Fr(\(1+)p Fp(")2619 5188 y Fh(2)2654 5211 y Fr(\))2686 5252 y Fs(:)1058 b Fw(\(1.11\))p eop %%Page: 18 22 18 21 bop -360 68 a Fw(18)718 b Fk(CHAPTER)34 b(1.)76 b(HARMONIC)33 b(ANAL)-8 b(YSIS)34 b(O)m(VER)f(FINITE)g(ABELIAN)h(GR)m (OUPS)-360 400 y Fw(The)i(follo)m(wing)c(equation)i(is)g(a)g(linear)f (com)m(bination)g(of)h(\(1.11\))g(with)g Fs(")2338 363 y Fr(2)2409 400 y Fw(=)d(1)j(and)h Fs(")2837 363 y Fr(2)2907 400 y Fw(=)c(1)p Fs(=)p Fw(2.)49 b(F)-8 b(or)34 b(an)m(y)h Fs(a)d Ft(\025)f Fw(0)-360 520 y(w)m(e)j(ha)m(v)m(e)567 699 y(1)p 565 743 52 4 v 565 834 a Fs(a)648 671 y Fm(X)643 887 y Fp(i)p Fo(2)p Fr([)p Fp(n)p Fr(])813 766 y Fs(I)856 781 y Fp(i)884 766 y Fw(\()p Fs(f)11 b Fw(\))22 b(+)1143 671 y Fm(X)1139 887 y Fp(i)p Fo(2)p Fr([)p Fp(n)p Fr(])1309 766 y Fs(I)1352 781 y Fp(i)1380 766 y Fw(\()p Fs(f)11 b Fw(\))1515 725 y Fr(4)p Fp(=)p Fr(3)1652 766 y Ft(\025)28 b Fw(4)1839 671 y Fm(X)1823 887 y Fp(x)p Fo(\022)p Fr([)p Fp(n)p Fr(])2016 626 y Fm(\022)2100 699 y Fw(1)p 2099 743 V 2099 834 a Fs(a)2182 766 y Fw(+)22 b(2)2329 725 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Fw(This)36 b(section)g(con)m (tains)g(the)g(pro)s(of)f(of)g(Lemma)g(1.5.2.)52 b(The)37 b(follo)m(wing)c(Lemmas)i(1.6.2)g(and)h(1.6.4)f(needed)j(in)-360 5280 y(the)j(pro)s(of)e(of)g(Lemma)g(1.5.2)h(w)m(ere)h(pro)m(v)m(en)h (b)m(y)f(Bec)m(kner)h([Bec75].)66 b(The)41 b(pro)s(ofs)f(giv)m(en)g (here)h(are)f(simpli\014ed)-360 5400 y(v)m(ersions)34 b(of)e(Bec)m(kner)i(pro)s(ofs.)p eop %%Page: 19 23 19 22 bop -180 69 a Fk(1.6.)76 b(BECKNER'S)33 b(LEMMAS)2836 b Fw(19)-180 400 y Fu(Lemma)37 b(1.6.1)49 b Fj(F)-7 b(or)34 b Fw(1)28 b Ft(\024)g Fs(p)f Ft(\024)i Fw(2)34 b Fj(and)h Fs(\013)28 b Ft(2)g Fq(R)678 629 y Fw(\(1)22 b(+)g(sin)16 b(2)p Fs(\013)q Fw(\))1170 581 y Fp(p=)p Fr(2)1303 629 y Fw(+)22 b(\(1)g Ft(\000)g Fw(sin)17 b(2)p Fs(\013)q Fw(\))1895 581 y Fp(p=)p Fr(2)p 678 674 1328 4 v 1317 765 a Fw(2)2043 697 y Ft(\025)2148 616 y Fm(\000)2194 697 y Fw(1)22 b(+)g(\(sin)16 b Fs(\013)q Fw(\))2638 655 y Fr(2)2677 697 y Fw(\()p Fs(p)22 b Ft(\000)h Fw(2\))2973 616 y Fm(\001)3018 638 y Fp(p=)p 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1721 y Fr(2)p Fp(k)2425 1606 y Fm(\001)2498 1686 y Fw(are)h(p)s(ositiv)m(e.)41 b(The)28 b(second)h(one)f(follo)m(ws)-180 1807 y(from)j(1)22 b(+)g Fs(xt)29 b Ft(\025)f Fw(\(1)22 b(+)g Fs(x)p Fw(\))743 1770 y Fp(t)773 1807 y Fs(;)49 b Fw(0)27 b Ft(\024)h Fs(x;)17 b(t)28 b Ft(\024)h Fw(1.)43 b(Hence)739 2029 y(\(1)22 b(+)g Fs(z)995 1993 y Fr(2)1057 2029 y Fw(+)g(2)p Fs(z)t Fw(\))1291 1993 y Fp(p=)p Fr(2)1424 2029 y Fw(+)g(\(1)g(+)g Fs(z)1778 1993 y Fr(2)1840 2029 y Ft(\000)h Fw(2)p Fs(z)t Fw(\))2076 1993 y Fp(p=)p Fr(2)p 739 2074 1448 4 v 1438 2165 a Fw(2)2224 2097 y Ft(\025)2329 2016 y Fm(\000)2375 2097 y Fw(1)f(+)g Fs(z)2593 2056 y Fr(2)2633 2097 y Fw(\()p Fs(p)g Ft(\000)h Fw(1\))2929 2016 y Fm(\001)2974 2038 y Fp(p=)p Fr(2)3084 2097 y Fs(:)-180 2372 y Fw(Divide)31 b(b)s(oth)i(sides)g(b)m(y)g(\(1)22 b(+)g Fs(z)986 2336 y Fr(2)1026 2372 y Fw(\))1064 2336 y Fp(p=)p Fr(2)1207 2372 y Fw(and)33 b(let)1593 2333 y Fp(z)1629 2309 y Fh(2)p 1547 2349 161 4 v 1547 2406 a Fr(1+)p Fp(z)1673 2387 y Fh(2)1746 2372 y Fw(=)27 b(\(sin)16 b Fs(\013)q Fw(\))2124 2336 y Fr(2)2163 2372 y Fw(.)1753 b Fi(\004)-180 2600 y Fu(Lemma)37 b(1.6.2)49 b Fj(F)-7 b(or)34 b(any)h Fs(x;)17 b(y)31 b Ft(2)d Fq(C)61 b Fj(and)34 b(any)h Fw(1)27 b Ft(\024)i Fs(p)e Ft(\024)h Fw(2)35 b Fj(we)f(have)487 2762 y Fm(\022)571 2835 y Ft(j)p Fs(x)22 b Fw(+)g Fs(y)826 2763 y Ft(p)p 908 2763 220 4 v 908 2835 a Fs(p)g Ft(\000)h Fw(1)p Ft(j)1156 2799 y Fr(2)1217 2835 y Fw(+)f Ft(j)p Fs(x)g Ft(\000)h Fs(y)1572 2763 y Ft(p)p 1654 2763 V 1654 2835 a Fs(p)f Ft(\000)h Fw(1)o Ft(j)1901 2799 y Fr(2)p 571 2880 1370 4 v 1231 2971 a Fw(2)1951 2762 y Fm(\023)2024 2781 y Fr(1)p Fp(=)p Fr(2)2162 2903 y Ft(\024)2267 2762 y Fm(\022)2350 2835 y Ft(j)p Fs(x)f Fw(+)g Fs(y)t Ft(j)2633 2799 y Fp(p)2694 2835 y Fw(+)g Ft(j)p Fs(x)g Ft(\000)h Fs(y)t Ft(j)3077 2799 y Fp(p)p 2350 2880 766 4 v 2708 2971 a Fw(2)3125 2762 y Fm(\023)3199 2785 y Fr(1)p Fp(=p)3325 2903 y Fs(:)419 b Fw(\(1.13\))-180 3180 y Fu(Pro)s(of)37 b(:)-180 3300 y Fw(W)-8 b(e)34 b(can)f(scale)h(inequalit)m(y)e(\(1.13\))h(so)g(that)h Fs(x)p 1527 3245 56 4 v(x)23 b Fw(+)g Fs(y)p 1756 3245 52 4 v 4 w(y)31 b Fw(=)e(1.)46 b(Let)33 b Fs(a)d Fw(=)f Fs(x)p 2478 3245 V(y)d Fw(+)p 2651 3245 56 4 v 23 w Fs(xy)36 b Fw(and)e(\(sin)16 b Fs(\013)q Fw(\))3256 3264 y Fr(2)3324 3300 y Fw(=)29 b Fs(y)p 3481 3245 52 4 v 4 w(y)s Fw(.)46 b(Note)33 b(that)-180 3421 y Fs(a)-129 3384 y Fr(2)-62 3421 y Ft(\024)28 b Fw(\(sin)17 b(2)p Fs(\013)q Fw(\))368 3384 y Fr(2)407 3421 y Fw(.)43 b(The)34 b(inequalit)m(y)d(b)s(ecomes) 791 3630 y Fm(\000)837 3711 y Fw(1)22 b(+)g(\(sin)16 b Fs(\013)q Fw(\))1281 3670 y Fr(2)1320 3711 y Fw(\()p Fs(p)22 b Ft(\000)h Fw(2\))1616 3630 y Fm(\001)1661 3652 y Fp(p=)p Fr(2)1799 3711 y Ft(\024)1904 3570 y Fm(\022)1988 3643 y Fw(\(1)e(+)h Fs(a)p Fw(\))2283 3607 y Fp(p=)p Fr(2)2416 3643 y Fw(+)g(\(1)g Ft(\000)g Fs(a)p Fw(\))2811 3607 y Fp(p=)p Fr(2)p 1988 3688 935 4 v 2430 3779 a Fw(2)2932 3570 y Fm(\023)3022 3711 y Fs(:)722 b Fw(\(1.14\))-180 3992 y(The)35 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b(=)1781 4643 y Fm(X)1783 4855 y Fp(x)p Fo(2)p Fp(A)1941 4738 y Fs(K)2024 4753 y Fp(T)2079 4738 y Fw(\()p Fs(x;)17 b(c)p Fw(\))p Fs(f)11 b Fw(\()p Fs(x)p Fw(\))p Fs(:)-180 5062 y Fw(F)-8 b(or)35 b(an)m(y)h Fs(B)k Fw(and)c Fs(f)43 b Ft(2)33 b Fq(C)748 5026 y Fp(A)p Fo(\002)p Fp(B)957 5062 y Fw(w)m(e)k(let)e Fs(T)14 b(f)43 b Fw(:)32 b Fq(C)1535 5026 y Fp(A)p Fo(\002)p Fp(B)1742 5062 y Ft(!)g Fq(C)1940 5026 y Fp(C)5 b Fo(\002)p Fp(B)2152 5062 y Fw(where)37 b(w)m(e)f(apply)f Fs(T)49 b Fw(to)35 b Fs(a)e Ft(!)f Fs(f)11 b Fw(\()p Fs(a;)17 b(b)p Fw(\))35 b(for)g(eac)m(h)i Fs(b)-180 5183 y Fw(separately)1255 5319 y(\()p Fs(T)14 b(f)d Fw(\)\()p Fs(c;)17 b(b)p Fw(\))27 b(=)1794 5224 y Fm(X)1797 5436 y Fp(x)p Fo(2)p Fp(A)1955 5319 y Fs(K)2038 5334 y Fp(T)2093 5319 y Fw(\()p Fs(x;)17 b(c)p Fw(\))p Fs(f)11 b Fw(\()p Fs(x;)17 b(b)p Fw(\))p eop %%Page: 20 24 20 23 bop -360 68 a Fw(20)718 b Fk(CHAPTER)34 b(1.)76 b(HARMONIC)33 b(ANAL)-8 b(YSIS)34 b(O)m(VER)f(FINITE)g(ABELIAN)h(GR)m (OUPS)-360 400 y Fw(Giv)m(en)c(t)m(w)m(o)h(op)s(erators)f Fs(T)588 415 y Fp(i)644 400 y Fw(:)e Fq(C)765 363 y Fp(A)818 373 y Fg(i)882 400 y Ft(!)f Fq(C)1075 363 y Fp(C)1126 373 y Fg(i)1162 400 y Fs(;)47 b(i)28 b Fw(=)g(1)p Fs(;)17 b Fw(2,)30 b(their)f Fu(pro)s(duct)i Fw(is)f(the)g(op)s(erator)g Fs(T)41 b Fw(:)28 b Fq(C)3117 363 y Fp(A)3170 372 y Fh(1)3211 363 y Fo(\002)p Fp(A)3319 372 y Fh(2)3385 400 y Ft(!)f Fq(C)3578 363 y Fp(C)3628 372 y Fh(1)3669 363 y Fo(\002)p Fp(C)3774 372 y Fh(2)3813 400 y Fw(,)-360 520 y Fs(T)50 b Fw(=)36 b Fs(T)-84 535 y Fr(1)-45 520 y Fs(T)12 535 y Fr(2)90 520 y Fw(where)i(w)m(e)h(apply)e Fs(T)855 535 y Fr(2)932 520 y Fw(for)g(eac)m(h)i Fs(a)d Ft(2)g Fs(A)1573 535 y Fr(1)1650 520 y Fw(separately)i(and)g(then)g Fs(T)2598 535 y Fr(1)2675 520 y Fw(for)f(eac)m(h)i Fs(d)c Ft(2)i Fs(C)3313 535 y Fr(2)3390 520 y Fw(separately)-8 b(.)-360 640 y(Note)33 b(that)f(the)h(k)m(ernel)g(of)f Fs(T)47 b Fw(is)923 860 y Fs(K)1006 875 y Fp(T)1061 779 y Fm(\000)1107 860 y Fw(\()p Fs(a;)17 b(b)p 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b(will)d(need)j(generalized)f(Mink)m (o)m(wski's)j(inequalit)m(y)-8 b(,)31 b(see)j([HLP34])f(p.)43 b(148.)-360 2733 y Fu(Lemma)37 b(1.6.3)49 b Fj(F)-7 b(or)34 b Fw(1)28 b Ft(\024)g Fs(p)f Ft(\024)i Fs(q)38 b Fj(and)d(any)f Fs(f)39 b Fw(:)28 b Fq(C)1527 2697 y Fp(A)p Fo(\002)p Fp(B)1729 2733 y Ft(!)f Fq(C)927 2952 y Ft(jj)17 b(jj)p Fs(f)11 b Fw(\()p Fs(x;)17 b(y)t Fw(\))p Ft(jj)1398 2967 y Fp(x)p Fr(:)p Fp(p)1510 2952 y Ft(jj)1566 2967 y Fp(y)r Fr(:)p Fp(q)1688 2952 y Ft(\024)28 b(jj)17 b(jj)p Fs(f)11 b Fw(\()p Fs(x;)17 b(y)t Fw(\))p Ft(jj)2264 2967 y Fp(y)r Fr(:)p Fp(q)2372 2952 y Ft(jj)2428 2967 y Fp(x)p Fr(:)p Fp(p)2526 2952 y Fs(:)-360 3171 y Fu(Lemma)37 b(1.6.4)49 b Fj(L)-5 b(et)35 b Fw(1)28 b Ft(\024)g Fs(p)f Ft(\024)i Fs(q)t Fj(.)44 b(L)-5 b(et)35 b Fs(T)1194 3186 y Fp(i)1250 3171 y Fw(:)28 b Fq(C)1370 3135 y Fp(A)1423 3145 y Fg(i)1488 3171 y Ft(!)f Fq(C)1681 3135 y Fp(C)1731 3145 y Fg(i)1768 3171 y Fs(;)51 b(i)28 b Fw(=)f(1)p Fs(;)17 b Fw(2)34 b Fj(b)-5 b(e)34 b(line)-5 b(ar)34 b(op)-5 b(er)g(ators.)44 b(If)34 b(for)g(any)h Fs(f)3535 3186 y Fp(i)3591 3171 y Ft(2)28 b Fq(C)3751 3135 y Fp(A)3804 3145 y Fg(i)1405 3391 y Ft(jj)p Fs(T)1518 3406 y Fp(i)1546 3391 y Fs(f)1594 3406 y Fp(i)1622 3391 y Ft(jj)1678 3406 y Fp(q)1743 3391 y Ft(\024)g(jj)p Fs(f)1952 3406 y Fp(i)1980 3391 y Ft(jj)2036 3406 y Fp(p)-360 3610 y Fj(then)35 b(for)f Fs(T)42 b Fw(=)27 b Fs(T)271 3625 y Fr(1)311 3610 y Fs(T)368 3625 y Fr(2)442 3610 y Fj(and)35 b(any)f Fs(f)39 b Ft(2)28 b Fq(C)1065 3574 y Fp(A)1118 3583 y Fh(1)1158 3574 y Fo(\002)p Fp(A)1266 3583 y Fh(2)1416 3829 y Ft(jj)p Fs(T)14 b(f)d Ft(jj)1658 3844 y Fp(q)1722 3829 y Ft(\024)28 b(jj)p Fs(f)11 b Ft(jj)1998 3844 y Fp(p)2037 3829 y Fs(:)-360 4048 y Fu(Pro)s(of)37 b(:)-360 4168 y Fw(Since)c(1)27 b Ft(\024)h Fs(p)g Ft(\024)g Fs(q)t Fw(,)33 b(w)m(e)g(can)g(use)h (Lemma)d(1.6.3)-277 4412 y Ft(jj)p Fw(\()p Fs(T)14 b(f)d Fw(\)\()p Fs(c;)17 b(d)p Fw(\))p Ft(jj)254 4427 y Fp(q)317 4412 y Fw(=)28 b Ft(jj)17 b(jj)p Fs(T)607 4427 y Fr(1)644 4412 y Fw(\()p Fs(T)739 4427 y Fr(2)779 4412 y Fs(f)11 b Fw(\)\()p Fs(c;)17 b(d)p Fw(\))p Ft(jj)1145 4427 y Fp(c)p Fr(:)p Fp(q)1248 4412 y Ft(jj)1304 4427 y Fp(d)p Fr(:)p Fp(q)1425 4412 y Ft(\024)28 b(jj)17 b(jj)p Fw(\()p Fs(T)1754 4427 y Fr(2)1792 4412 y Fs(f)11 b Fw(\)\()p Fs(a;)17 b(d)p Fw(\))p Ft(jj)2167 4427 y Fp(a)p Fr(:)p Fp(p)2279 4412 y Ft(jj)2335 4427 y Fp(d)p Fr(:)p Fp(q)2456 4412 y Ft(\024)1647 4558 y(jj)g(jj)p Fw(\()p Fs(T)1871 4573 y Fr(2)1909 4558 y Fs(f)11 b Fw(\)\()p Fs(a;)17 b(d)p Fw(\))p Ft(jj)2284 4573 y Fp(d)p Fr(:)p Fp(q)2393 4558 y Ft(jj)2449 4573 y Fp(a)p Fr(:)p Fp(p)2573 4558 y Ft(\024)28 b(jj)17 b(jj)p Fs(f)11 b Fw(\()p Fs(a;)17 b(b)p Fw(\))p Ft(jj)3134 4573 y Fp(b)p Fr(:)p Fp(p)3237 4558 y Ft(jj)3293 4573 y Fp(a)p Fr(:)p Fp(p)3417 4558 y Fw(=)28 b Ft(jj)p Fs(f)11 b Ft(jj)3692 4573 y Fp(p)3730 4558 y Fs(:)3763 4777 y Fi(\004)-360 4968 y Fw(Lemma)31 b(1.6.4)h(can)h(b)s(e)g(easily)f(extended)i(for)e(the)h(pro)s(duct)g (of)f(more)g(than)h(2)f(op)s(erators.)43 b(No)m(w)34 b(w)m(e)f(can)g(\014nally)-360 5088 y(pro)m(v)m(e)h(Lemma)d(1.5.2.)-360 5280 y Fu(Pro)s(of)37 b(of)h(Lemma)f(1.5.2)-360 5400 y Fw(T)-8 b(ak)m(e)28 b(the)e(op)s(erator)g Fs(T)475 5415 y Fp(n;")602 5400 y Fw(:)i(\()p Fq(Z)764 5364 y Fp(n)764 5425 y Fr(2)836 5400 y Ft(!)f Fq(C)20 b Fw(\))34 b Ft(!)27 b Fw(\()p Fq(Z)1335 5364 y Fp(n)1335 5425 y Fr(2)1407 5400 y Ft(!)g Fq(C)20 b Fw(\))33 b(de\014ned)28 b(b)m(y)f(\(1.9\).)41 b(T)-8 b(ak)m(e)27 b(an)m(y)h(function)e Fs(f)38 b Fw(:)28 b Fq(Z)3393 5415 y Fr(2)3457 5400 y Ft(!)g Fq(C)52 b Fw(and)p eop %%Page: 21 25 21 24 bop -180 69 a Fk(1.6.)76 b(BECKNER'S)33 b(LEMMAS)2836 b Fw(21)-180 400 y(let)26 b Fs(x)i Fw(=)159 373 y Fm(b)142 400 y Fs(f)10 b Fw(\(0\),)28 b Fs(y)j Fw(=)580 373 y Fm(b)562 400 y Fs(f)11 b Fw(\(1\).)41 b(Then)28 b Fs(f)11 b Fw(\(0\))27 b(=)g Fs(x)10 b Fw(+)g Fs(y)t Fw(,)27 b Fs(f)11 b Fw(\(1\))27 b(=)h Fs(x)10 b Ft(\000)g Fs(y)30 b Fw(and)c(\()p Fs(T)2457 415 y Fr(1)p Fp(;")2549 400 y Fs(f)11 b Fw(\)\(0\))27 b(=)h Fs(x)10 b Fw(+)g Fs("y)t Fw(,)26 b(\()p Fs(T)3299 415 y Fr(1)p Fp(;")3391 400 y Fs(f)11 b Fw(\)\(1\))27 b(=)h Fs(x)10 b Ft(\000)g Fs("y)t Fw(.)-180 520 y(T)-8 b(aking)32 b Fs(p)c Fw(=)f(1)22 b(+)g Fs(")540 484 y Fr(2)612 520 y Fw(in)32 b(Lemma)f(1.6.2)h(w)m(e)i (obtain)d(that)h Fs(T)2022 535 y Fr(1)p Fp(;")2142 520 y Fw(:)c Fs(L)2263 484 y Fr(1+)p Fp(")2386 460 y Fh(2)2424 520 y Fw(\()p Fq(Z)2531 535 y Fr(2)2568 520 y Fw(\))g Ft(!)f Fs(L)2827 484 y Fr(2)2867 520 y Fw(\()p Fq(Z)2974 535 y Fr(2)3011 520 y Fw(\))32 b(is)g(a)h(norm)e(1)i(op)s(erator.)-80 688 y(No)m(w)g(w)m(e)h(will)d(sho)m(w)j(that)f Fs(T)982 703 y Fp(n;")1114 688 y Fw(is)f(pro)s(duct)i(of)e Fs(n)h Fw(copies)g(of)g Fs(T)2238 703 y Fr(1)p Fp(;")2363 688 y Fw(and)g(hence)h(b)m(y)g(Lemma)d(1.6.4)i(has)g(norm)f(1)-180 808 y(as)h(an)f(op)s(erator)g(from)f Fs(L)764 772 y Fr(1+)p Fp(")887 748 y Fh(2)926 808 y Fw(\()p Fq(Z)1033 772 y Fp(n)1033 833 y Fr(2)1078 808 y Fw(\))h(to)g Fs(L)1333 772 y Fr(2)1373 808 y Fw(\()p Fq(Z)1480 772 y Fp(n)1480 833 y Fr(2)1524 808 y Fw(\).)-80 990 y(Let)46 b Fs(F)171 1005 y Fp(n)268 990 y Fw(:)k Fq(C)411 954 y Fc(Z)467 930 y Fg(n)467 974 y Fh(2)559 990 y Ft(!)f Fq(C)777 936 y Ff(b)774 954 y Fc(Z)830 930 y Fg(n)830 974 y Fh(2)917 990 y Fw(b)s(e)d(the)g(F)-8 b(ourier)45 b(transform)g(op)s(erator.)83 b(Let)46 b Fs(U)2781 1005 y Fp(n)2878 990 y Fw(:)k Fq(C)3024 936 y Ff(b)3021 954 y Fc(Z)3077 930 y Fg(n)3077 974 y Fh(2)3169 990 y Ft(!)f Fq(C)3387 936 y Ff(b)3384 954 y Fc(Z)3440 930 y Fg(n)3440 974 y Fh(2)3527 990 y Fw(ha)m(v)m(e)e(k)m (ernel)-180 1110 y Fs(K)-97 1125 y Fp(U)-38 1110 y Fw(\()p Fs(x;)17 b(c)p Fw(\))28 b(=)f(\()p Fs(x)h Fw(=)g Fs(c)p Fw(\))p Fs(")661 1074 y Fo(j)p Fp(x)p Fo(j)743 1110 y Fw(.)43 b(Clearly)27 b Fs(F)1210 1125 y Fp(n)1286 1110 y Fw(is)h(the)h(pro)s(duct)g(of)g Fs(n)f Fw(copies)h(of)g Fs(F)2555 1125 y Fr(1)2623 1110 y Fw(and)g Fs(U)2875 1125 y Fp(n)2950 1110 y Fw(is)g(the)g(pro)s(duct)g(of)f Fs(n)h Fw(copies)-180 1231 y(of)j Fs(U)-3 1246 y Fr(1)36 1231 y Fw(.)44 b(No)m(w)33 b(since)f Fs(T)624 1246 y Fp(n)699 1231 y Fw(=)c Fs(F)880 1194 y Fo(\000)p Fr(1)866 1255 y Fp(n)995 1231 y Ft(\016)21 b Fs(U)1132 1246 y Fp(n)1201 1231 y Ft(\016)g Fs(F)1335 1246 y Fp(n)1415 1231 y Fw(w)m(e)33 b(ha)m(v)m(e)g(using)f(\(1.15\))g(that)g Fs(T)2587 1246 y Fp(n)2666 1231 y Fw(is)g(the)g(pro)s(duct)h(of)f Fs(n)g Fw(copies)h(of)e Fs(T)3953 1246 y Fr(1)3993 1231 y Fw(.)-180 1351 y Fi(\004)p eop %%Page: 22 26 22 25 bop -360 68 a Fw(22)718 b Fk(CHAPTER)34 b(1.)76 b(HARMONIC)33 b(ANAL)-8 b(YSIS)34 b(O)m(VER)f(FINITE)g(ABELIAN)h(GR)m (OUPS)p eop %%Page: 23 27 23 26 bop -180 1064 a Fy(Chapter)78 b(2)-180 1479 y(Harmonic)f (Analysis)e(o)-6 b(v)g(er)76 b(Lo)6 b(cally)-180 1728 y(Compact)78 b(Ab)6 b(elian)77 b(Groups)-180 2243 y Fn(2.1)161 b(In)l(tro)t(duction)-180 2462 y Fw(W)-8 b(e)37 b(will)d(only)h(use)i (harmonic)e(analysis)h(on)g Fq(R)1577 2426 y Fp(n)1630 2462 y Fw(,)h Fq(Z)1763 2426 y Fp(n)1844 2462 y Fw(and)f Fq(T)2100 2426 y Fp(n)2184 2462 y Fw(=)e Fq(R)2360 2426 y Fp(n)2413 2462 y Fs(=)p Fq(Z)2531 2426 y Fp(n)2575 2462 y Fw(.)54 b(It)37 b(migh)m(t)d(b)s(e)j(useful)f(to)g(men)m(tion)f (a)-180 2582 y(more)g(general)g(setting.)53 b(A)36 b Fu(lo)s(cally)j(compact)h(ab)s(elian)h(\(LCA\))35 b Fw(group)g(is)h(a)f (lo)s(cally)e(compact)j(Hausdor\013)-180 2703 y(top)s(ological)e(space) k(with)f(group)g(op)s(erations)f(whic)m(h)i(are)f(con)m(tin)m(uous)h (i.e.)57 b(the)38 b(mappings)e(+)f(:)h Fs(G)25 b Ft(\002)h Fs(G)36 b Ft(!)f Fs(G)-180 2823 y Fw(\(where)f Fs(G)22 b Ft(\002)g Fs(G)33 b Fw(has)g(Cartesian)f(pro)s(duct)h(top)s(ology\))e (and)i Ft(\000)28 b Fw(:)g Fs(G)f Ft(!)h Fs(G)k Fw(are)h(con)m(tin)m (uous.)-80 2943 y(On)24 b(suc)m(h)i(a)e(group)g(there)h(alw)m(a)m(ys)g (exists)g(unique)f(\(up)h(to)f(m)m(ultiplicativ)m(e)d(factor\))i(Borel) h(measure)g Fs(\026)g Fw(in)m(v)-5 b(arian)m(t)-180 3064 y(under)41 b(the)f(group)f(op)s(erations)g(i.e.)65 b(for)39 b(ev)m(ery)j(measurable)d(set)i Fs(E)k Fw(and)40 b Fs(x)h Ft(2)f Fs(G)p Fw(,)i Fs(\026)p Fw(\()p Fs(E)6 b Fw(\))39 b(=)h Fs(\026)p Fw(\()p Fs(E)33 b Fw(+)27 b Fs(x)p Fw(\).)65 b(The)-180 3184 y(measure)35 b(is)g(called)f(the)h Fu(Haar)41 b(measure)p Fw(.)51 b(The)36 b(measure)f(of)f(a)h(compact)g(set)g(is)g (\014nite)g(and)g(the)g(measure)g(of)-180 3304 y(an)30 b(op)s(en)g(set)g(is)f(p)s(ositiv)m(e.)42 b(The)31 b(in)m(tegral)d (corresp)s(onding)h(to)h(the)g(Haar)f(measure)h(is)g(called)e(the)i (Haar)f(in)m(tegral.)-80 3425 y(A)d Fu(c)m(haracter)h Fw(of)g Fs(G)f Fw(is)h(a)f(con)m(tin)m(uous)i(homomorphisms)c(from)i Fs(G)g Fw(to)h(the)g(m)m(ultiplicativ)m(e)c(group)k(of)f(complex)-180 3545 y(n)m(um)m(b)s(ers)33 b(of)g(mo)s(dulus)e(1.)43 b(The)34 b(group)1333 3520 y Fm(b)1314 3545 y Fs(G)f Fw(of)f(all)e(c)m(haracters)k(of)e Fs(G)h Fw(with)f(top)s(ology)f (generated)i(b)m(y)600 3749 y Fs(U)666 3764 y Fp(C)q(;")801 3749 y Fw(=)27 b Ft(f)p Fs(\037)h Ft(2)1156 3724 y Fm(b)1137 3749 y Fs(G)p Fw(;)17 b Ft(8)p Fs(x)28 b Ft(2)g Fs(C)35 b Fw(:)28 b Ft(j)p Fs(\037)p Fw(\()p Fs(x)p Fw(\))22 b Ft(\000)h Fw(1)p Ft(j)k Fs(<)g(";)17 b Ft(g)p Fs(;)49 b(C)29 b Ft(\000)22 b Fw(compact)p Fs(;)17 b(")27 b(>)h Fw(0)-180 3954 y(\(and)i(their)g(translates\))g(is)f(a)h(lo)s(cally)d (compact)j(ab)s(elian)e(group)i(called)f(the)i Fu(dual)j(group)d Fw(of)e Fs(G)p Fw(.)43 b(P)m(on)m(try)m(agin's)-180 4084 y Fu(dualit)m(y)34 b(theorem)c Fw(sa)m(ys)i(that)f(the)g(dual)f(of)1554 4059 y Fm(b)1535 4084 y Fs(G)h Fw(is)f(isomorphic)f(\(as)i(a)f(top)s (ological)d(group\))k(to)f Fs(G)p Fw(.)43 b(Moreo)m(v)m(er)32 b(if)-180 4214 y Fs(G)g Fw(is)h(discrete)g(then)627 4189 y Fm(b)608 4214 y Fs(G)f Fw(is)h(compact)f(and)g(vice)h(v)m(ersa.)-80 4335 y(The)g Fu(F)-9 b(ourier)37 b(T)-9 b(ransform)32 b Fw(of)g Fs(f)39 b Ft(2)28 b Fs(L)1406 4299 y Fr(1)1445 4335 y Fw(\()p Fs(G)p Fw(\))33 b(is)f(de\014ned)i(as)1451 4561 y Fm(b)1433 4588 y Fs(f)10 b Fw(\()p Fs(\037)p Fw(\))28 b(=)1760 4452 y Fm(Z)1815 4678 y Fp(G)1891 4588 y Fs(f)11 b Fw(\()p Fs(g)t Fw(\))p 2077 4501 188 4 v Fs(\037)p Fw(\()p Fs(g)t Fw(\))j Fs(dg)t(:)-180 4868 y Fw(If)33 b Fs(f)43 b Fw(is)32 b(con)m(tin)m(uous)h(and)803 4842 y Fm(b)785 4868 y Fs(f)38 b Ft(2)28 b Fs(L)1031 4832 y Fr(1)1071 4868 y Fw(\()1128 4843 y Fm(b)1109 4868 y Fs(G)p Fw(\))k(then)i(w)m(e)f(ha)m(v)m(e)h(the)f Fu(in)m(v)m(ersion)j (form)m(ula)1098 5121 y Fs(f)11 b Fw(\()p Fs(x)p Fw(\))28 b(=)1420 4985 y Fm(Z)1489 5194 y Ff(b)1475 5211 y Fp(G)1569 5094 y Fm(b)1551 5121 y Fs(f)11 b Fw(\()p Fs(x)p Fw(\))p Fs(\037)p Fw(\()p Fs(x)p Fw(\))17 b Fs(d\037;)114 b Fw(for)32 b(all)e Fs(x)f Ft(2)f Fs(G)-180 5400 y Fw(where)34 b Fs(d\037)e Fw(is)g(an)h(appropriate)e(normalization)e(of)j(the)h(Haar)f (measure)h(on)2678 5375 y Fm(b)2659 5400 y Fs(G)p Fw(.)1871 5649 y(23)p eop %%Page: 24 28 24 27 bop -360 68 a Fw(24)57 b Fk(CHAPTER)34 b(2.)76 b(HARMONIC)33 b(ANAL)-8 b(YSIS)34 b(O)m(VER)f(LOCALL)-8 b(Y)33 b(COMP)-8 b(A)m(CT)34 b(ABELIAN)g(GR)m(OUPS)-260 400 y Fw(If)e Fs(f)39 b Ft(2)28 b Fs(L)84 363 y Fr(1)123 400 y Fw(\()p Fs(G)p Fw(\))22 b Ft(\\)h Fs(L)453 363 y Fr(2)493 400 y Fw(\()p Fs(G)p Fw(\))32 b(then)918 373 y Fm(b)900 400 y Fs(f)39 b Ft(2)28 b Fs(L)1147 363 y Fr(2)1187 400 y Fw(\()1243 374 y Fm(b)1225 400 y Fs(G)o Fw(\))33 b(and)g(w)m(e)g(ha)m(v)m(e)h(the)f Fu(Planc)m(herel)j(form)m (ula)1466 633 y Ft(jj)p Fs(f)11 b Ft(jj)1637 648 y Fr(2)1702 633 y Fw(=)28 b Ft(jj)1879 606 y Fm(b)1862 633 y Fs(f)10 b Ft(jj)1976 648 y Fr(2)-360 855 y Fu(Remark)37 b(2)49 b Fw(If)34 b Fs(G)g Fw(is)g(compact)g(w)m(e)h(usually)f(normalize)e (the)j(Haar)f(measure)g(on)h Fs(G)f Fw(so)g(that)h Fs(\026)p Fw(\()p Fs(G)p Fw(\))30 b(=)g(1.)49 b(This)-360 986 y(normalization)38 b(mak)m(es)43 b(the)f(c)m(haracters)h(orthonormal.)69 b(The)43 b(normalization)38 b(of)j(the)i(Haar)e(measure)h(on)3782 960 y Fm(b)3763 986 y Fs(G)-360 1106 y Fw(\(whic)m(h)33 b(is)f(discrete\))h(for)f(whic)m(h)h(the)g(in)m(v)m(ersion)g(form)m (ula)e(holds)h(is)g Fs(\026)p Fw(\()p Fs(x)p Fw(\))c(=)f(1)p Fs(;)49 b(x)28 b Ft(2)h Fs(G)p Fw(.)-260 1227 y(Note)36 b(that)f(the)i(Haar)e(measures)i(de\014ned)g(for)e(a)h(\014nite)f (group)h Fs(G)g Fw(in)f(section)h(1)g(corresp)s(ond)g(to)g(viewing)f Fs(G)-360 1347 y Fw(as)e(a)f(compact)g(group)h(and)716 1322 y Fm(b)697 1347 y Fs(G)g Fw(as)g(a)f(discrete)h(group.)-360 1753 y Fn(2.2)161 b(F)-13 b(ourier)53 b(transform)g(o)l(v)l(er)f Fb(T)1755 1701 y Fs(n)-360 1973 y Fw(Let)36 b Fq(T)-119 1937 y Fp(n)-35 1973 y Fw(=)d Fq(R)140 1937 y Fp(n)193 1973 y Fs(=)p Fq(Z)311 1937 y Fp(n)391 1973 y Fw(with)i(the)h(standard) g(Leb)s(esgue)i(measure)e(and)g(in)m(tegral.)51 b(Tw)m(o)37 b(functions)f Fs(f)5 b(;)17 b(g)36 b Fw(:)e Fq(T)3552 1937 y Fp(n)3635 1973 y Ft(!)f Fq(C)-360 2093 y Fw(are)g(equiv)-5 b(alen)m(t)32 b Fs(f)38 b Ft(\030)28 b Fs(g)36 b Fw(if)c(they)h (di\013er)f(in)g(a)g(set)i(of)e(p)s(oin)m(ts)g(of)g(measure)h(0.)-260 2215 y(Let)g Fs(L)-19 2178 y Fr(2)21 2215 y Fw(\()p Fq(T)122 2178 y Fp(n)172 2215 y Fw(\))h(b)s(e)g(the)g(space)g(of)f(measurable)g (functions)h Fq(T)1916 2178 y Fp(n)1995 2215 y Ft(!)29 b Fq(C)60 b Fw(suc)m(h)35 b(that)2663 2134 y Fm(R)2710 2249 y Fc(T)2760 2230 y Fg(n)2818 2215 y Ft(j)p Fs(f)11 b Fw(\()p Fs(x)p Fw(\))p Ft(j)3064 2178 y Fr(2)3119 2215 y Fs(d\026)29 b(<)g Ft(1)k Fw(factored)-360 2335 y(b)m(y)g(the)g(equiv) -5 b(alence)33 b(relation)e Ft(\030)p Fw(.)44 b(The)33 b(space)h Fs(L)1491 2299 y Fr(2)1531 2335 y Fw(\()p Fq(T)1632 2299 y Fp(n)1682 2335 y Fw(\))f(with)f(inner)g(pro)s(duct)1270 2605 y Ft(h)p Fs(f)5 b(;)17 b(g)t Ft(i)26 b Fw(=)1626 2470 y Fm(Z)1681 2695 y Fp(T)1753 2605 y Fs(f)11 b Fw(\()p Fs(t)p Fw(\))p 1923 2519 162 4 v Fs(g)t Fw(\()p Fs(t)p Fw(\))16 b Fs(d\026)-360 2874 y Fw(is)32 b(a)g(Hilb)s(ert)f(space.)-260 2995 y(Let)d Fs(L)-24 2959 y Fr(2)15 2995 y Fw(\()p Fq(Z)122 2959 y Fp(n)167 2995 y Fw(\))f(b)s(e)i(the)f(space)h(of)e(functions)h Fs(f)38 b Fw(:)28 b Fq(Z)1512 2959 y Fp(n)1584 2995 y Ft(!)f Fq(C)54 b Fw(suc)m(h)29 b(that)2233 2921 y Fm(P)2338 3024 y Fp(z)s Fo(2)p Fc(Z)2471 3006 y Fg(n)2528 2995 y Ft(j)p Fs(f)11 b Fw(\()p Fs(z)t Fw(\))p Ft(j)2768 2959 y Fr(2)2835 2995 y Fs(<)28 b Ft(1)p Fs(:)f Fw(The)i(space)g Fs(L)3611 2959 y Fr(2)3651 2995 y Fw(\()p Fq(Z)3758 2959 y Fp(n)3802 2995 y Fw(\))-360 3116 y(with)j(inner)g(pro)s(duct)1289 3250 y Ft(h)p Fs(f)5 b(;)17 b(g)t Ft(i)26 b Fw(=)1658 3155 y Fm(X)1645 3367 y Fp(z)s Fo(2)p Fc(Z)1778 3348 y Fg(n)1831 3250 y Fs(f)11 b Fw(\()p Fs(z)t Fw(\))p 2015 3163 176 4 v Fs(g)t Fw(\()p Fs(z)t Fw(\))-360 3515 y(is)32 b(a)g(Hilb)s(ert)f(space.)-360 3739 y Fu(Theorem)37 b(2.2.1)49 b Fj(Inner)34 b(pr)-5 b(o)g(duct)35 b(and)f(metric)h(satisfy)-215 3946 y Ft(\017)49 b(j)p Fw(\()p Fs(f)5 b(;)17 b(g)t Fw(\))p Ft(j)26 b(\024)i(jj)p Fs(f)11 b Ft(jj)466 3961 y Fr(2)526 3946 y Ft(\001)22 b(jj)p Fs(g)t Ft(jj)739 3961 y Fr(2)811 3946 y Fj(\(the)35 b(Cauchy-Schwarz)f(ine)-5 b(quality\))-215 4153 y Ft(\017)49 b(jj)p Fs(f)32 b Fw(+)22 b Fs(g)t Ft(jj)225 4168 y Fr(2)291 4153 y Ft(\024)28 b(jj)p Fs(f)11 b Ft(jj)567 4168 y Fr(2)627 4153 y Fw(+)22 b Ft(jj)p Fs(g)t Ft(jj)888 4168 y Fr(2)960 4153 y Fj(\(triangle)35 b(ine)-5 b(quality\))-360 4376 y Fu(Theorem)37 b(2.2.2)49 b Fj(The)34 b(char)-5 b(acters)35 b(of)f Fq(T)1239 4340 y Fp(n)1325 4376 y Fj(ar)-5 b(e)1068 4598 y Ft(f)p Fs(\037)1179 4613 y Fp(z)1218 4598 y Fw(\()p Fs(x)p Fw(\))28 b(=)g(exp)q(\(2)p Fs(\031)t(iz)1858 4557 y Fp(T)1914 4598 y Fs(x)p Fw(\);)17 b Fs(z)32 b Ft(2)c Fq(Z)2291 4557 y Fp(n)2336 4598 y Ft(g)p Fs(:)-360 4820 y Fj(They)35 b(form)f(an)g(orthonormal)g(b)-5 b(asis)34 b(of)h Fs(L)1234 4784 y Fr(2)1274 4820 y Fw(\()p Fq(T)1375 4784 y Fp(n)1425 4820 y Fw(\))g Fj(i.e.)44 b(any)35 b(function)g Fs(f)j Ft(2)28 b Fs(L)2494 4784 y Fr(2)2534 4820 y Fw(\()p Fq(T)2635 4784 y Fp(n)2685 4820 y Fw(\))35 b Fj(c)-5 b(an)34 b(b)-5 b(e)35 b(written)g(as)1396 5060 y Fs(f)j Fw(=)1599 4965 y Fm(X)1586 5177 y Fp(z)s Fo(2)p Fc(Z)1719 5158 y Fg(n)1790 5034 y Fm(b)1772 5060 y Fs(f)11 b Fw(\()p Fs(z)t Fw(\))p Fs(\037)2017 5075 y Fp(z)2057 5060 y Fs(:)-260 5400 y Fw(The)-42 5374 y Fm(b)-60 5400 y Fs(f)g Fw(\()p Fs(z)t Fw(\))28 b Ft(2)g Fq(C)59 b Fw(are)32 b(called)g(the)h(F)-8 b(ourier)31 b(co)s(e\016cien)m(ts,)1834 5374 y Fm(b)1816 5400 y Fs(f)11 b Fw(\()p Fs(z)t Fw(\))28 b(=)f Ft(h)p Fs(f)5 b(;)17 b(\037)2328 5415 y Fp(z)2368 5400 y Ft(i)p Fs(:)p eop %%Page: 25 29 25 28 bop -180 68 a Fk(2.3.)76 b(EXP)-8 b(ANDER)33 b(GRAPHS)f(CONSTR)m (UCTION)2029 b Fw(25)-180 400 y Fu(Theorem)37 b(2.2.3)h(\(Riesz-Fisc)m (her\))47 b Fj(The)34 b(map)g Fs(L)1827 363 y Fr(2)1867 400 y Fw(\()p Fq(T)1968 363 y Fp(n)2018 400 y Fw(\))28 b Ft(!)2112 363 y Fo(^)2198 400 y Fs(L)2264 363 y Fr(2)2303 400 y Fw(\()p Fq(Z)2410 363 y Fp(n)2455 400 y Fw(\))35 b Fj(is)f(line)-5 b(ar,)34 b(bije)-5 b(ctive)34 b(map.)-180 584 y Fw(Moreo)m(v)m(er)g(w)m(e)g(ha)m(v)m(e)-35 778 y Ft(\017)49 b(h)p Fs(f)5 b(;)17 b(g)t Ft(i)26 b Fw(=)i Ft(h)477 752 y Fm(b)459 778 y Fs(f)10 b(;)17 b Fm(b)-55 b Fs(g)s Ft(i)33 b Fw(\(P)m(arsev)-5 b(al's)33 b(iden)m(tit)m(y\))-35 989 y Ft(\017)49 b(jj)p Fs(f)11 b Ft(jj)235 1004 y Fr(2)300 989 y Fw(=)28 b Ft(jj)477 963 y Fm(b)460 989 y Fs(f)10 b Ft(jj)574 1004 y Fr(2)645 989 y Fw(\(the)33 b(Planc)m(herel)g(form)m (ula\))-35 1201 y Ft(\017)49 b(jj)137 1174 y Fm(b)120 1201 y Fs(f)10 b Ft(jj)234 1216 y Fo(1)336 1201 y Ft(\024)28 b(jj)p Fs(f)11 b Ft(jj)612 1216 y Fr(1)-180 1395 y Fw(and)33 b(hence)h(the)f(F)-8 b(ourier)31 b(transform)h(is)g(an)g(isometry)g(of) g Fs(L)2043 1358 y Fr(2)2083 1395 y Fw(\()p Fq(T)2184 1358 y Fp(n)2234 1395 y Fw(\))h(and)f Fs(L)2560 1358 y Fr(2)2600 1395 y Fw(\()p Fq(Z)2707 1410 y Fp(n)2752 1395 y Fw(\).)-80 1562 y(A)g(nice)h(application)d(of)i(the)h(Planc)m (herel)f(form)m(ula)f(is)h(computing)g(the)h(sum)1679 1717 y Fo(1)1643 1747 y Fm(X)1648 1956 y Fp(n)p Fr(=1)1838 1774 y Fw(1)p 1813 1819 98 4 v 1813 1910 a Fs(n)1871 1881 y Fr(2)1948 1841 y Fw(=)2062 1774 y Fs(\031)2121 1738 y Fr(2)p 2062 1819 99 4 v 2087 1910 a Fw(6)2170 1841 y Fs(:)-180 2161 y Fw(Consider)j Fs(f)11 b Fw(\()p Fs(x)p Fw(\))32 b(=)g Fs(x)k Fw(on)f Fq(T)852 2125 y Fr(1)895 2161 y Fw(.)52 b(Then)36 b Ft(jj)p Fs(f)11 b Ft(jj)1402 2125 y Fr(2)1402 2186 y(2)1473 2161 y Fw(=)32 b(1)p Fs(=)p Fw(3,)1808 2135 y Fm(b)1790 2161 y Fs(f)11 b Fw(\(0\))32 b(=)g(1)p Fs(=)p Fw(2)j(and)2506 2135 y Fm(b)2488 2161 y Fs(f)11 b Fw(\()p Fs(n)p Fw(\))33 b(=)f Fs(i=)p Fw(\(2)p Fs(\031)t(n)p Fw(\))j(for)g Fs(n)d Ft(6)p Fw(=)h(0.)51 b(No)m(w)36 b(use)-180 2281 y(the)d(Planc)m(herel)g(form)m (ula.)-180 2613 y Fn(2.3)161 b(Expander)53 b(Graphs)g(Construction)-180 2832 y Fw(The)25 b(\014rst)f(expander)h(graph)f(construction)g(is)f (due)h(to)g(Margulis)e([Mar73].)40 b(W)-8 b(e)25 b(are)e(going)g(to)g (sho)m(w)i(an)f(expander)-180 2952 y(graph)39 b(construction)h(due)g (to)f(Gabb)s(er)g(and)g(Galil)d([GG79)o(].)64 b(The)40 b(construction)g(can)f(b)s(e)h(analyzed)f(without)-180 3072 y(harmonic)21 b(analysis,)i(using)f(linear)f(algebra)f(\(eigen)m (v)-5 b(alues\))22 b([JM87)q(].)40 b(Expanders)24 b(with)d(m)m(uc)m(h)i (b)s(etter)f(expansion)-180 3193 y(ha)m(v)m(e)34 b(b)s(een)f (constructed)h(in)e([LPS88,)h(Mar88].)-180 3408 y Fu(De\014nition)j (2.3.1)49 b Fw(A)e(bipartite)d(graph)i Fs(G)51 b Fw(=)g(\()p Fs(A)31 b Ft([)h Fs(B)5 b(;)17 b(E)6 b Fw(\))46 b(is)g(\()p Fs(n;)17 b(d;)g(\013)q Fw(\))45 b Fu(expander)i Fw(if)e Ft(j)p Fs(A)p Ft(j)51 b Fw(=)g Ft(j)p Fs(B)5 b Ft(j)50 b Fw(=)h Fs(n)p Fw(,)-180 3528 y Ft(j)p Fs(E)6 b Ft(j)27 b(\024)h Fs(nd)33 b Fw(and)f(for)g(ev)m(ery)j Fs(X)g Ft(\022)28 b Fs(A)1362 3717 y(N)10 b Fw(\()p Fs(X)e Fw(\))28 b Ft(\025)1748 3577 y Fm(\022)1821 3717 y Fw(1)22 b(+)g Fs(\013)2063 3650 y Ft(j)p 2091 3570 89 4 v Fs(X)7 b Ft(j)p 2063 3694 145 4 v 2106 3785 a Fs(n)2217 3577 y Fm(\023)2307 3717 y Ft(j)p Fs(X)h Ft(j)p Fs(:)-180 3953 y Fw(Explicit)33 b(expanders)k(of)d(b)s(ounded)i(degree)f(ha)m(v)m(e)i (a)d(great)h(n)m(um)m(b)s(er)g(of)f(applications)f(in)h(the)h(Theory)h (of)e(Com-)-180 4074 y(puting.)42 b(They)30 b(are)f(used)i(in)d(the)i (log)16 b Fs(n)p Fw(-depth)30 b(sorting)e(net)m(w)m(ork)j(of)e(Ajtai,)f (Koml\023)-49 b(os)28 b(and)h(Szemer)m(\023)-46 b(edi)29 b([AKS83],)-180 4194 y(in)35 b(extraction)g(of)g(random)g(bits)g(from)f (w)m(eak)j(random)d(sources)j([AKS87,)f(CW89,)g(IZ89)o(],)g(or)f(in)g (explicit)f(con-)-180 4314 y(struction)e(of)h(fault)e(toleran)m(t)h (net)m(w)m(orks)i([A)m(C88)q(].)-180 4530 y Fu(De\014nition)i(2.3.2)49 b Fw(Let)42 b Fs(G)883 4545 y Fp(m)992 4530 y Fw(b)s(e)g(graph)g(with)g Fs(A;)17 b(B)48 b Fw(=)c Fq(Z)2080 4494 y Fr(2)2080 4554 y Fp(m)2186 4530 y Fw(and)e(\()p Fs(x;)17 b(y)t Fw(\))43 b Ft(2)h Fs(A)e Fw(connected)i(to)d Ft(f)p Fs(\033)3578 4545 y Fp(i)3607 4530 y Fw(\()p Fs(x;)17 b(y)t Fw(\);)g Fs(i)43 b Ft(2)-180 4650 y(f)p Fw(0)p Fs(;)17 b(:::;)g Fw(4)p Ft(gg)31 b Fw(where)1010 4859 y Fs(\033)1065 4874 y Fr(0)1105 4859 y Fw(\()p Fs(x;)17 b(y)t Fw(\))27 b(=)g(\()p Fs(x;)17 b(y)t Fw(\))1010 5004 y Fs(\033)1065 5019 y Fr(1)1105 5004 y Fw(\()p Fs(x;)g(y)t Fw(\))27 b(=)g(\()p Fs(x)c Fw(+)f Fs(y)t(;)17 b(y)t Fw(\))p Fs(;)80 b(\033)2024 5019 y Fr(2)2064 5004 y Fw(\()p Fs(x;)17 b(y)t Fw(\))26 b(=)i(\()p Fs(x)22 b Fw(+)g Fs(y)k Fw(+)c(1)p Fs(;)17 b(y)t Fw(\))1010 5150 y Fs(\033)1065 5165 y Fr(3)1105 5150 y Fw(\()p Fs(x;)g(y)t Fw(\))27 b(=)g(\()p Fs(x;)17 b(x)23 b Fw(+)f Fs(y)t Fw(\))p Fs(;)81 b(\033)2028 5165 y Fr(4)2068 5150 y Fw(\()p Fs(x;)17 b(y)t Fw(\))27 b(=)g(\()p Fs(x;)17 b(x)23 b Fw(+)f Fs(y)j Fw(+)d(1\))p -14 5281 3869 4 v -14 5446 4 166 v 46 5397 a Fu(Theorem)38 b(2.3.3)49 b Fs(G)878 5412 y Fp(m)979 5397 y Fj(is)35 b(an)f Fw(\()p Fs(m)1346 5361 y Fr(2)1386 5397 y Fs(;)17 b Fw(5)p Fs(;)g Fw(\(2)k Ft(\000)1731 5314 y(p)p 1814 5314 49 4 v 83 x Fw(3)o(\))p Fs(=)p Fw(4\))35 b Fj(exp)-5 b(ander.)p 3851 5446 4 166 v -14 5449 3869 4 v eop %%Page: 26 30 26 29 bop -360 68 a Fw(26)57 b Fk(CHAPTER)34 b(2.)76 b(HARMONIC)33 b(ANAL)-8 b(YSIS)34 b(O)m(VER)f(LOCALL)-8 b(Y)33 b(COMP)-8 b(A)m(CT)34 b(ABELIAN)g(GR)m(OUPS)-360 400 y Fu(Pro)s(of)j(:)-360 520 y Fw(W)-8 b(e)33 b(need)h(to)e(sho)m(w)i (for)e(an)m(y)h Fs(X)i Ft(\022)28 b Fq(Z)1020 484 y Fr(2)1020 545 y Fp(m)877 807 y Ft(j)p Fs(\033)960 822 y Fr(1)1000 807 y Fw(\()p Fs(X)8 b Fw(\))22 b Ft([)g(\001)17 b(\001)g(\001)k([)h Fs(\033)1557 822 y Fr(4)1597 807 y Fw(\()p Fs(X)8 b Fw(\))22 b Ft(\000)g Fs(X)8 b Ft(j)28 b(\025)g Fs(\013)2206 740 y Ft(j)p Fs(X)8 b Ft(j)21 b(\001)g(j)p 2449 660 89 4 v Fs(X)8 b Ft(j)p 2206 784 361 4 v 2323 875 a Fs(m)2408 847 y Fr(2)2576 807 y Fs(:)-360 1061 y Fw(There)36 b(is)d(a)h(natural)f (mapping)g(b)s(et)m(w)m(een)k(the)d(subsets)j(of)c Fq(Z)1921 1025 y Fr(2)1921 1086 y Fp(m)2020 1061 y Fw(and)h(the)h(subsets)h(of)e (the)h(2-dimensional)c(torus)-360 1182 y(with)h(sides)h(of)f(length)g Fs(m)h Fw(\(i.e.)43 b Fq(R)895 1145 y Fr(2)941 1182 y Fs(=m)p Fq(Z)1144 1145 y Fr(2)1181 1182 y Fw(\),)32 b(where)i(\()p Fs(i;)17 b(j)6 b Fw(\))32 b(corresp)s(onds)i(to)e Fi(\003)p Fw(\()p Fs(i;)17 b(j)6 b Fw(\))28 b(=)g Ft(h)p Fs(i;)17 b(i)22 b Fw(+)g(1\))g Ft(\002)h(h)p Fs(j;)17 b(j)28 b Fw(+)22 b(1\).)-260 1302 y(Let)45 b Fs(\034)-31 1317 y Fr(1)54 1302 y Fw(b)s(e)h(a)f(linear)f(transformation)f(on)i Fq(R)1469 1266 y Fr(2)1515 1302 y Fs(=m)p Fq(Z)1718 1266 y Fr(2)1755 1302 y Fw(,)j Fs(\034)1872 1317 y Fr(1)1912 1302 y Fw(\()p Fs(x;)17 b(y)t Fw(\))49 b(=)g(\()p Fs(x)31 b Fw(+)g Fs(y)t(;)17 b(y)t Fw(\).)80 b(Under)46 b Fs(\034)3187 1317 y Fr(1)3272 1302 y Fw(the)g Fi(\003)p Fw(\()p Fs(i;)17 b(j)6 b Fw(\))45 b(is)-360 1422 y(transformed)31 b(to)g(a)g (parallelogram)c(one)k(half)f(of)h(whic)m(h)h(lies)e(in)h Fi(\003)p Fw(\()p Fs(i)20 b Fw(+)f Fs(j;)e(j)6 b Fw(\))31 b(and)h(the)f(other)h(in)e Fi(\003)p Fw(\()p Fs(i)21 b Fw(+)e Fs(j)25 b Fw(+)20 b(1)p Fs(;)d(j)6 b Fw(\).)-360 1543 y(Hence)39 b(the)f(nonempt)m(y)h(squares)g(in)e Fs(\034)1080 1558 y Fr(1)1120 1543 y Fw(\()p Fs(X)1247 1507 y Fo(0)1270 1543 y Fw(\))26 b Ft(\000)g Fs(X)1526 1507 y Fo(0)1587 1543 y Fw(corresp)s(ond)39 b(to)e(the)h(elemen)m(ts)g (in)f(\()p Fs(\033)3002 1558 y Fr(1)3042 1543 y Fw(\()p Fs(X)8 b Fw(\))26 b Ft([)g Fs(\033)3380 1558 y Fr(2)3420 1543 y Fw(\()p Fs(X)8 b Fw(\)\))25 b Ft(\000)h Fs(X)-360 1663 y Fw(where)34 b Fs(X)11 1627 y Fo(0)62 1663 y Ft(\022)28 b Fq(R)233 1627 y Fr(2)278 1663 y Fs(=)p Fq(Z)396 1627 y Fr(2)396 1688 y Fp(m)492 1663 y Fw(corresp)s(onds)34 b(to)f Fs(X)i Ft(\022)28 b Fq(Z)1437 1627 y Fr(2)1437 1688 y Fp(m)1501 1663 y Fw(.)44 b(Th)m(us)893 1882 y Ft(j)p Fs(\033)976 1897 y Fr(1)1016 1882 y Fw(\()p Fs(X)8 b Fw(\))21 b Ft([)i Fs(\033)1346 1897 y Fr(2)1386 1882 y Fw(\()p Fs(X)8 b Fw(\))22 b Ft(\000)g Fs(X)8 b Ft(j)27 b(\025)i Fs(\026)p Fw(\()p Fs(\034)2061 1897 y Fr(1)2100 1882 y Fw(\()p Fs(X)2227 1841 y Fo(0)2250 1882 y Fw(\))22 b Ft(\000)h Fs(X)2499 1841 y Fo(0)2522 1882 y Fw(\))p Fs(:)-360 2101 y Fw(Similarly)29 b(for)j Fs(\034)246 2116 y Fr(2)285 2101 y Fw(\()p Fs(x;)17 b(y)t Fw(\))27 b(=)h(\()p Fs(x;)17 b(x)22 b Fw(+)g Fs(y)t Fw(\),)32 b Ft(j)p Fs(\033)1187 2116 y Fr(3)1227 2101 y Fw(\()p Fs(X)8 b Fw(\))21 b Ft([)i Fs(\033)1557 2116 y Fr(4)1597 2101 y Fw(\()p Fs(X)8 b Fw(\))22 b Ft(\000)g Fs(X)8 b Ft(j)27 b(\025)i Fs(\026)p Fw(\()p Fs(\034)2272 2116 y Fr(2)2311 2101 y Fw(\()p 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629 y Fw(+)22 b Fs(T)1365 644 y Fr(2)1405 629 y Fw(\))p Fu(1)1499 644 y Fp(X)1566 629 y Ft(i)28 b(\024)g Fw(2)p Fs(\026)p Fw(\()p Fs(X)8 b Fw(\)\()p Fs(\014)e(=)p Fw(2)20 b(+)i(\(1)g Ft(\000)h Fs(\014)6 b(=)p Fw(2\))p Fs(\026)p Fw(\()p Fs(X)i Fw(\)\))-180 858 y(whic)m(h)33 b(pro)m(v)m(es)h(\(2.4\))e(for)g Fs(\013)d Fw(=)e(1)22 b Ft(\000)h Fs(\014)6 b(=)p Fw(2)27 b(=)g(\(3)22 b Ft(\000)1645 776 y(p)p 1728 776 49 4 v 82 x Fw(2\))p Fs(=)p Fw(4)27 b Fs(<)g Fw(\(4)22 b Ft(\000)h Fw(2)2301 776 y Ft(p)p 2384 776 V 82 x Fw(3)o(\))p Fs(=)p Fw(4.)-80 982 y(T)-8 b(o)32 b(\014x)h(the)g(pro)s(of)f(note)h(that)f (if)g(for)g(ev)m(ery)i(measurable)e Fs(X)8 b Fw(,)893 1211 y Fs(\026)p Fw(\()p Fs(\034)1043 1170 y Fr(2)1032 1236 y(1)1082 1211 y Fw(\()p Fs(X)g Fw(\))22 b Ft(\000)h Fs(X)8 b Fw(\))22 b(+)g Fs(\026)p Fw(\()p Fs(\034)1766 1170 y Fr(2)1755 1236 y(2)1805 1211 y Fw(\()p Fs(X)8 b Fw(\))22 b Ft(\000)h Fs(X)8 b Fw(\)\))27 b Ft(\025)h Fw(4)p Fs(\013)q(\026)p Fw(\()p Fs(X)8 b Fw(\))p Fs(\026)p Fw(\()p 2822 1131 89 4 v Fs(X)e Fw(\))872 b(\(2.6\))-180 1441 y(then)39 b(also)f(\(2.2\).)62 b(The)40 b(pro)s(of)e(of)g(\(2.6\)) g(follo)m(ws)g(exactly)h(the)g(same)g(steps)h(as)f(the)g(pro)s(of)f(of) h(\(2.2\),)h(using)e(an)-180 1561 y(estimate)32 b(on)g(the)h(Ra)m (yleigh)f(quotien)m(t)g(of)g Fs(T)1483 1525 y Fo(0)1534 1561 y Fw(=)c Fs(T)1695 1576 y Fr(1)1756 1561 y Ft(\016)22 b Fs(T)1885 1576 y Fr(1)1947 1561 y Fw(+)g Fs(T)2102 1576 y Fr(2)2164 1561 y Ft(\016)g Fs(T)2293 1576 y Fr(2)2332 1561 y Fw(.)1584 b Fi(\004)-180 1913 y Fn(2.4)161 b(The)53 b(Ra)l(yleigh)i(quotien)l(t)d(of)i(Op)t(erators)-180 2139 y Fw(In)31 b(this)f(section)g(w)m(e)i(compute)e(the)h(Ra)m(yleigh) e(quotien)m(t)i(of)f(the)h(op)s(erator)f Fs(T)2658 2154 y Fr(1)2715 2139 y Fw(+)18 b Fs(T)2866 2154 y Fr(2)2936 2139 y Fw(and)30 b Fs(T)3180 2154 y Fr(1)3238 2139 y Ft(\016)17 b Fs(T)3362 2154 y Fr(1)3420 2139 y Fw(+)g Fs(T)3570 2154 y Fr(2)3628 2139 y Ft(\016)g Fs(T)3752 2154 y Fr(2)3822 2139 y Fw(from)-180 2259 y(the)33 b(section)g(2.3)f (follo)m(wing)e(the)j(pro)s(of)e(of)i(Gabb)s(er)f(and)g(Galil)e([GG79)n (].)-80 2383 y(Giv)m(en)i(a)g(b)s(ounded)i(linear)d(op)s(erator)h Fs(T)46 b Fw(its)32 b Fu(sp)s(ectral)37 b(norm)32 b Fw(is)1213 2634 y Ft(jj)p Fs(T)14 b Ft(jj)1396 2649 y Fr(2)1461 2634 y Fw(=)28 b(sup)1728 2523 y Fm(n)1795 2634 y Ft(jj)p Fs(T)14 b(x)p Ft(jj)2033 2649 y Fr(2)2088 2634 y Fw(;)33 b Ft(jj)p Fs(x)p Ft(jj)2315 2649 y Fr(2)2381 2634 y Fw(=)28 b(1)2534 2523 y Fm(o)2600 2634 y Fs(;)-180 2885 y Fw(its)k Fu(sp)s(ectral)37 b(radius)c Fw(is)1469 3015 y Fs(\032)p Fw(\()p Fs(T)14 b Fw(\))28 b(=)52 b(lim)1797 3075 y Fp(n)p Fo(!1)1998 3015 y Ft(jj)p Fs(T)2125 2974 y Fp(n)2171 3015 y Ft(jj)2227 2964 y Fr(1)p Fp(=n)2227 3040 y Fr(2)2344 3015 y Fs(;)-180 3227 y Fw(and)33 b(its)f Fu(Ra)m(yleigh)k(quotien)m(t) 31 b Fw(is)1186 3479 y Fs(r)s Fw(\()p Fs(T)14 b Fw(\))27 b(=)g(sup)1674 3368 y Fm(n)1740 3479 y Ft(jh)p Fs(T)14 b(x;)j(x)p Ft(ij)g Fw(;)32 b Ft(jj)p Fs(x)p Ft(jj)2342 3494 y Fr(2)2408 3479 y Fw(=)c(1)2561 3368 y Fm(o)2627 3479 y Fs(:)-180 3730 y Fw(The)34 b(Cauc)m(h)m(y-Sc)m(h)m(w)m(arz)i (inequalit)m(y)31 b(yields)1468 3959 y Fs(\032)p Fw(\()p Fs(T)14 b Fw(\))28 b Ft(\024)g Fs(r)s Fw(\()p Fs(T)14 b Fw(\))27 b Ft(\024)h(jj)p Fs(T)14 b Ft(jj)2307 3974 y Fr(2)2345 3959 y Fs(:)-180 4189 y Fw(If)33 b Fs(T)46 b Fw(is)32 b(self-adjoin)m(t)f(then)i Fs(\032)p Fw(\()p Fs(T)14 b Fw(\))27 b(=)h Fs(r)s Fw(\()p Fs(T)14 b Fw(\))27 b(=)g Ft(jj)p Fs(T)14 b Ft(jj)1678 4204 y Fr(2)1716 4189 y Fw(.)-80 4393 y(The)38 b(linear)f(op)s(erator)g Fs(T)52 b Fw(on)37 b Fs(L)1115 4357 y Fr(2)1155 4393 y Fw(\()p Fq(T)1256 4357 y Fp(n)1306 4393 y Fw(\))h(giv)m(es)g(us)h(a)f(linear)e (op)s(erator)2534 4368 y Fm(b)2518 4393 y Fs(T)51 b Fw(=)36 b Fs(F)k Ft(\016)25 b Fs(T)40 b Ft(\016)25 b Fs(F)3165 4357 y Fo(\000)p Fr(1)3297 4393 y Fw(on)38 b Fs(L)3504 4357 y Fr(2)3544 4393 y Fw(\()p Fq(Z)3651 4357 y Fp(n)3695 4393 y Fw(\))g(where)-180 4523 y Fs(F)49 b Fw(is)34 b(the)h(F)-8 b(ourier)34 b(transform)g(op)s(erator.)50 b(By)36 b(P)m(arsev)-5 b(al's)35 b(iden)m(tit)m(y)g(the)g(op)s(erators)3023 4498 y Fm(b)3007 4523 y Fs(T)49 b Fw(and)35 b Fs(T)48 b Fw(ha)m(v)m(e)37 b(the)e(same)-180 4644 y(sp)s(ectral)41 b(norm,)i(radius)e(and)h(Ra)m(yleigh)e(quotien)m(t.)70 b(In)42 b(our)g(application)d(it)h(will)f(turn)j(out)f(to)g(b)s(e)h (easier)f(to)-180 4764 y(analyze)33 b(the)g(op)s(erator)746 4739 y Fm(b)730 4764 y Fs(T)14 b Fw(.)-180 5003 y Fu(Lemma)37 b(2.4.1)49 b Fj(L)-5 b(et)32 b Fs(A)f Fj(b)-5 b(e)30 b(an)h(inte)-5 b(gr)g(al)30 b Fs(n)13 b Ft(\002)g Fs(n)33 b Fj(matrix)d(with)h(determinant)f Fw(1)h Fj(and)f Fw(\()p Fs(T)14 b(f)d Fw(\)\()p Fs(x)p Fw(\))27 b(=)h Fs(f)11 b Fw(\()p Fs(Ax)p Fw(\))31 b Fj(a)g(line)-5 b(ar)-180 5123 y(op)g(er)g(ator)35 b(on)f Fs(L)409 5087 y Fr(2)449 5123 y Fw(\()p Fq(T)550 5087 y Fp(n)600 5123 y Fw(\))p Fj(.)45 b(Then)1486 5364 y Fw(\()1540 5339 y Fm(b)1524 5364 y Fs(T)1613 5338 y Fm(b)1595 5364 y Fs(f)10 b Fw(\)\()p Fs(x)p Fw(\))28 b(=)1972 5338 y Fm(b)1954 5364 y Fs(f)11 b Fw(\()p Fs(A)2124 5323 y Fo(\000)p Fp(T)2234 5364 y Fs(x)p Fw(\))p Fs(:)1465 b Fw(\(2.7\))p eop %%Page: 28 32 28 31 bop -360 68 a Fw(28)57 b Fk(CHAPTER)34 b(2.)76 b(HARMONIC)33 b(ANAL)-8 b(YSIS)34 b(O)m(VER)f(LOCALL)-8 b(Y)33 b(COMP)-8 b(A)m(CT)34 b(ABELIAN)g(GR)m(OUPS)-360 400 y Fu(Pro)s(of)j(:)-360 520 y Fw(Since)-89 495 y Fm(b)-105 520 y Fs(T)46 b Fw(is)32 b(linear,)f(it)h(is)g(enough)h(to)f(see)i (\(2.7\))e(for)1603 494 y Fm(b)1585 520 y Fs(f)38 b Fw(=)28 b Fu(1)1831 535 y Fp(a)-115 791 y Fw(\()-61 766 y Fm(b)-77 791 y Fs(T)14 b Fu(1)50 806 y Fp(a)91 791 y Fw(\)\()p Fs(x)p Fw(\))28 b(=)392 656 y Fm(Z)447 881 y Fc(T)497 862 y Fg(n)555 791 y Fs(e)600 750 y Fo(\000)p Fr(2)p Fp(\031)r(ix)797 727 y Fg(T)845 750 y Fp(z)885 791 y Fs(T)14 b Fw(\()p Fs(e)1039 750 y Fr(2)p Fp(\031)r(ia)1178 727 y Fg(T)1227 750 y Fp(z)1267 791 y Fw(\))j Fs(d)1373 750 y Fp(n)1419 791 y Fs(z)33 b Fw(=)1600 656 y Fm(Z)1656 881 y Fc(T)1706 862 y Fg(n)1763 791 y Fs(e)1808 750 y Fr(2)p Fp(\031)r(i)p Fr(\()p Fp(a)1974 727 y Fg(T)2024 750 y Fp(A)p Fo(\000)p Fp(x)2172 727 y Fg(T)2220 750 y Fr(\))p Fp(z)2304 791 y Fs(d)2355 750 y Fp(n)2402 791 y Fs(z)f Fw(=)c(\()p Fs(x)g Fw(=)f Fs(A)2880 750 y Fp(T)2936 791 y Fs(a)p Fw(\))g(=)h Fu(1)3212 806 y Fp(a)3254 791 y Fw(\()p Fs(A)3365 750 y Fo(\000)p Fp(T)3475 791 y Fs(x)p Fw(\))p Fs(:)3763 1060 y Fi(\004)-260 1323 y Fw(Let)39 b Fs(X)46 b Fw(b)s(e)39 b(the)h(subspace)g(of)f Fs(L)959 1287 y Fr(2)999 1323 y Fw(\()p Fq(T)1100 1287 y Fr(2)1142 1323 y Fw(\))g(con)m(taining)f(the)h(functions)g(satisfying)2759 1296 y Fm(b)2742 1323 y Fs(f)10 b Fw(\(0\))38 b(=)3077 1242 y Fm(R)3160 1323 y Fs(f)28 b(d)3287 1287 y Fr(2)3326 1323 y Fs(x)39 b Fw(=)f(0.)62 b(Our)-360 1443 y(goal)30 b(no)m(w)h(is)g(to)g(compute)g(the)h(Ra)m(yleigh)e(quotien)m(t)i(of)e (the)i(op)s(erator)f Fs(T)41 b Fw(=)28 b Fs(T)2534 1458 y Fr(1)2593 1443 y Fw(+)19 b Fs(T)2745 1458 y Fr(2)2816 1443 y Fw(\(de\014ned)32 b(in)f(\(2.4\)\))f(on)i(the)-360 1563 y(space)i Fs(X)8 b Fw(.)43 b(Recall)31 b(that)925 1687 y(\()p Fs(T)14 b(f)d Fw(\)\()p Fs(x;)17 b(y)t Fw(\))26 b(=)i Fs(f)11 b Fw(\()p Fs(x)22 b Ft(\000)h Fs(y)t(;)17 b(y)t Fw(\))j(+)i Fs(f)11 b Fw(\()p Fs(x;)17 b(y)25 b Ft(\000)d Fs(x)p Fw(\))p Fs(:)-360 1865 y Fw(Hence)34 b(b)m(y)f(Lemma)f(2.4.1)940 1988 y(\()994 1963 y Fm(b)978 1988 y Fs(T)1067 1962 y Fm(b)1049 1988 y Fs(f)11 b Fw(\)\()p Fs(x;)17 b(y)t Fw(\))26 b(=)1521 1962 y Fm(b)1503 1988 y Fs(f)11 b Fw(\()p Fs(x;)17 b(x)22 b Fw(+)g Fs(y)t Fw(\))g(+)2102 1962 y Fm(b)2084 1988 y Fs(f)10 b Fw(\()p Fs(x)23 b Fw(+)f Fs(y)t(;)17 b(y)t Fw(\))-360 2176 y(where)-52 2151 y Fm(b)-68 2176 y Fs(T)57 b Fw(is)43 b(op)s(erator)f(on)h(the)h(space)g Fs(Y)65 b Fw(whic)m(h)43 b(is)g(a)g(subspace)i(of)d Fs(L)2371 2140 y Fr(2)2411 2176 y Fw(\()p Fq(Z)2518 2140 y Fr(2)2555 2176 y Fw(\))h(consisting)g(of)f(functions)h(with)-360 2296 y Fs(f)11 b Fw(\(0\))27 b(=)h(0.)-260 2489 y(Consider)34 b(an)f(undirected)i(graph)e Fs(H)1126 2504 y Fr(1)1199 2489 y Fw(with)g(v)m(ertices)i Fq(Z)1845 2452 y Fr(2)1905 2489 y Ft(\000)23 b(f)p Fw(0)p Ft(g)33 b Fw(and)h(\()p Fs(a;)17 b(b)p Fw(\))34 b(b)s(eing)f(connected)i(to)f(\()p Fs(a)23 b Fw(+)f Fs(b;)17 b(b)p Fw(\))-360 2619 y(and)33 b(\()p Fs(a;)17 b(a)22 b Fw(+)g Fs(b)p Fw(\).)44 b(W)-8 b(e)33 b(de\014ne)h(a)e(function)g Fs(r)j Fw(on)e(undirected)g(lo)s (cally)d(\014nite)i(graphs)h(so)g(that)f Fs(r)s Fw(\()p Fs(H)3264 2634 y Fr(1)3303 2619 y Fw(\))c(=)f Fs(r)s Fw(\()3573 2594 y Fm(b)3557 2619 y Fs(T)13 b Fw(\).)1693 2863 y Fa(u)186 b(u)1457 3099 y(u)1457 3335 y(u)1693 3571 y(u)g(u)2165 3099 y(u)2165 3335 y(u)1622 3028 y(u)1787 2957 y(u)2000 3028 y(u)2071 3193 y(u)2000 3406 y(u)1834 3477 y(u)1622 3406 y(u)1551 3240 y(u)p 1453 3099 7 3 v 1456 3096 V 1459 3093 V 1462 3090 V 1465 3087 V 1467 3085 V 1470 3082 V 1473 3079 V 1476 3076 V 1479 3073 V 1482 3070 V 1484 3068 V 1487 3065 V 1490 3062 V 1493 3059 V 1496 3056 V 1499 3053 V 1501 3051 V 1504 3048 V 1507 3045 V 1510 3042 V 1513 3039 V 1516 3036 V 1518 3034 V 1521 3031 V 1524 3028 V 1527 3025 V 1530 3022 V 1533 3019 V 1535 3017 V 1538 3014 V 1541 3011 V 1544 3008 V 1547 3005 V 1550 3002 V 1552 3000 V 1555 2997 V 1558 2994 V 1561 2991 V 1564 2988 V 1567 2985 V 1569 2983 V 1572 2980 V 1575 2977 V 1578 2974 V 1581 2971 V 1584 2968 V 1586 2966 V 1589 2963 V 1592 2960 V 1595 2957 V 1598 2954 V 1601 2951 V 1603 2949 V 1606 2946 V 1609 2943 V 1612 2940 V 1615 2937 V 1618 2934 V 1620 2932 V 1623 2929 V 1626 2926 V 1629 2923 V 1632 2920 V 1635 2917 V 1637 2915 V 1640 2912 V 1643 2909 V 1646 2906 V 1649 2903 V 1652 2900 V 1654 2897 V 1657 2895 V 1660 2892 V 1663 2889 V 1666 2886 V 1669 2883 V 1671 2880 V 1674 2878 V 1677 2875 V 1680 2872 V 1683 2869 V 1686 2866 V 1688 2863 V 1693 2866 237 7 v 1926 2865 7 3 v 1928 2868 V 1931 2871 V 1934 2874 V 1937 2877 V 1940 2880 V 1943 2882 V 1945 2885 V 1948 2888 V 1951 2891 V 1954 2894 V 1957 2897 V 1960 2899 V 1962 2902 V 1965 2905 V 1968 2908 V 1971 2911 V 1974 2914 V 1977 2916 V 1980 2919 V 1982 2922 V 1985 2925 V 1988 2928 V 1991 2931 V 1994 2933 V 1997 2936 V 1999 2939 V 2002 2942 V 2005 2945 V 2008 2948 V 2011 2950 V 2014 2953 V 2016 2956 V 2019 2959 V 2022 2962 V 2025 2965 V 2028 2967 V 2031 2970 V 2033 2973 V 2036 2976 V 2039 2979 V 2042 2982 V 2045 2984 V 2048 2987 V 2050 2990 V 2053 2993 V 2056 2996 V 2059 2999 V 2062 3001 V 2065 3004 V 2067 3007 V 2070 3010 V 2073 3013 V 2076 3016 V 2079 3018 V 2082 3021 V 2084 3024 V 2087 3027 V 2090 3030 V 2093 3033 V 2096 3035 V 2099 3038 V 2101 3041 V 2104 3044 V 2107 3047 V 2110 3050 V 2113 3052 V 2116 3055 V 2118 3058 V 2121 3061 V 2124 3064 V 2127 3067 V 2130 3069 V 2133 3072 V 2135 3075 V 2138 3078 V 2141 3081 V 2144 3084 V 2147 3086 V 2150 3089 V 2152 3092 V 2155 3095 V 2158 3098 V 2161 3101 V 2162 3335 7 237 v 2162 3338 7 3 v 2159 3341 V 2156 3343 V 2153 3346 V 2151 3349 V 2148 3352 V 2145 3355 V 2142 3358 V 2139 3360 V 2136 3363 V 2134 3366 V 2131 3369 V 2128 3372 V 2125 3375 V 2122 3377 V 2119 3380 V 2117 3383 V 2114 3386 V 2111 3389 V 2108 3392 V 2105 3394 V 2102 3397 V 2100 3400 V 2097 3403 V 2094 3406 V 2091 3409 V 2088 3411 V 2085 3414 V 2083 3417 V 2080 3420 V 2077 3423 V 2074 3426 V 2071 3429 V 2068 3431 V 2065 3434 V 2063 3437 V 2060 3440 V 2057 3443 V 2054 3446 V 2051 3448 V 2048 3451 V 2046 3454 V 2043 3457 V 2040 3460 V 2037 3463 V 2034 3465 V 2031 3468 V 2029 3471 V 2026 3474 V 2023 3477 V 2020 3480 V 2017 3482 V 2014 3485 V 2012 3488 V 2009 3491 V 2006 3494 V 2003 3497 V 2000 3499 V 1997 3502 V 1995 3505 V 1992 3508 V 1989 3511 V 1986 3514 V 1983 3516 V 1980 3519 V 1978 3522 V 1975 3525 V 1972 3528 V 1969 3531 V 1966 3533 V 1963 3536 V 1961 3539 V 1958 3542 V 1955 3545 V 1952 3548 V 1949 3550 V 1946 3553 V 1944 3556 V 1941 3559 V 1938 3562 V 1935 3565 V 1932 3567 V 1929 3570 V 1927 3573 V 1693 3575 237 7 v 1689 3571 7 3 v 1687 3568 V 1684 3566 V 1681 3563 V 1678 3560 V 1675 3557 V 1672 3554 V 1670 3551 V 1667 3549 V 1664 3546 V 1661 3543 V 1658 3540 V 1655 3537 V 1653 3534 V 1650 3532 V 1647 3529 V 1644 3526 V 1641 3523 V 1638 3520 V 1636 3517 V 1633 3514 V 1630 3512 V 1627 3509 V 1624 3506 V 1621 3503 V 1619 3500 V 1616 3497 V 1613 3495 V 1610 3492 V 1607 3489 V 1604 3486 V 1602 3483 V 1599 3480 V 1596 3478 V 1593 3475 V 1590 3472 V 1587 3469 V 1585 3466 V 1582 3463 V 1579 3461 V 1576 3458 V 1573 3455 V 1570 3452 V 1568 3449 V 1565 3446 V 1562 3444 V 1559 3441 V 1556 3438 V 1553 3435 V 1551 3432 V 1548 3429 V 1545 3427 V 1542 3424 V 1539 3421 V 1536 3418 V 1534 3415 V 1531 3412 V 1528 3410 V 1525 3407 V 1522 3404 V 1519 3401 V 1517 3398 V 1514 3395 V 1511 3393 V 1508 3390 V 1505 3387 V 1502 3384 V 1500 3381 V 1497 3378 V 1494 3376 V 1491 3373 V 1488 3370 V 1485 3367 V 1483 3364 V 1480 3361 V 1477 3359 V 1474 3356 V 1471 3353 V 1468 3350 V 1466 3347 V 1463 3344 V 1460 3342 V 1457 3339 V 1454 3336 V 1453 3335 7 237 v 1457 3102 7 7 v 1463 3099 V 1469 3096 V 1476 3094 V 1482 3091 V 1488 3088 V 1495 3085 V 1501 3082 V 1508 3079 V 1514 3077 V 1520 3074 V 1527 3071 V 1533 3068 V 1539 3065 V 1546 3062 V 1552 3060 V 1559 3057 V 1565 3054 V 1571 3051 V 1578 3048 V 1584 3045 V 1590 3043 V 1597 3040 V 1603 3037 V 1610 3034 V 1616 3031 V 1622 3031 V 1628 3028 V 1635 3026 V 1641 3023 V 1647 3020 V 1654 3017 V 1660 3014 V 1667 3011 V 1673 3009 V 1679 3006 V 1686 3003 V 1692 3000 V 1698 2997 V 1705 2994 V 1711 2992 V 1718 2989 V 1724 2986 V 1730 2983 V 1737 2980 V 1743 2977 V 1749 2975 V 1756 2972 V 1762 2969 V 1769 2966 V 1775 2963 V 1781 2960 V 1787 2960 5 7 v 1791 2958 V 1796 2955 V 1800 2952 V 1804 2949 V 1808 2946 V 1813 2943 V 1817 2940 V 1821 2938 V 1826 2935 V 1830 2932 V 1834 2929 V 1838 2926 V 1843 2923 V 1847 2921 V 1851 2918 V 1855 2915 V 1860 2912 V 1864 2909 V 1868 2906 V 1872 2904 V 1877 2901 V 1881 2898 V 1885 2895 V 1889 2892 V 1894 2889 V 1898 2887 V 1902 2884 V 1906 2881 V 1911 2878 V 1915 2875 V 1919 2872 V 1923 2870 V 1928 2867 V 1926 2869 7 7 v 1928 2875 V 1931 2882 V 1934 2888 V 1937 2894 V 1940 2901 V 1943 2907 V 1945 2914 V 1948 2920 V 1951 2926 V 1954 2933 V 1957 2939 V 1960 2945 V 1962 2952 V 1965 2958 V 1968 2965 V 1971 2971 V 1974 2977 V 1977 2984 V 1980 2990 V 1982 2996 V 1985 3003 V 1988 3009 V 1991 3016 V 1994 3022 V 1997 3028 V 1997 3034 V 1999 3041 V 2002 3047 V 2005 3053 V 2008 3060 V 2011 3066 V 2014 3073 V 2016 3079 V 2019 3085 V 2022 3092 V 2025 3098 V 2028 3104 V 2031 3111 V 2033 3117 V 2036 3124 V 2039 3130 V 2042 3136 V 2045 3143 V 2048 3149 V 2050 3155 V 2053 3162 V 2056 3168 V 2059 3175 V 2062 3181 V 2065 3187 V 2067 3194 V 2067 3197 7 5 v 2070 3202 V 2073 3206 V 2076 3210 V 2079 3214 V 2082 3219 V 2084 3223 V 2087 3227 V 2090 3231 V 2093 3236 V 2096 3240 V 2099 3244 V 2101 3249 V 2104 3253 V 2107 3257 V 2110 3261 V 2113 3266 V 2116 3270 V 2118 3274 V 2121 3278 V 2124 3283 V 2127 3287 V 2130 3291 V 2133 3295 V 2135 3300 V 2138 3304 V 2141 3308 V 2144 3312 V 2147 3317 V 2150 3321 V 2152 3325 V 2155 3329 V 2158 3334 V 2161 3338 V 2159 3338 7 7 v 2152 3341 V 2146 3344 V 2140 3347 V 2133 3350 V 2127 3352 V 2121 3355 V 2114 3358 V 2108 3361 V 2101 3364 V 2095 3367 V 2089 3369 V 2082 3372 V 2076 3375 V 2070 3378 V 2063 3381 V 2057 3384 V 2050 3386 V 2044 3389 V 2038 3392 V 2031 3395 V 2025 3398 V 2018 3401 V 2012 3403 V 2006 3406 V 1999 3409 V 1993 3409 V 1987 3412 V 1981 3415 V 1974 3418 V 1968 3420 V 1962 3423 V 1955 3426 V 1949 3429 V 1942 3432 V 1936 3435 V 1930 3437 V 1923 3440 V 1917 3443 V 1911 3446 V 1904 3449 V 1898 3452 V 1891 3455 V 1885 3457 V 1879 3460 V 1872 3463 V 1866 3466 V 1860 3469 V 1853 3472 V 1847 3474 V 1840 3477 V 1834 3480 V 1830 3480 5 7 v 1826 3483 V 1822 3486 V 1817 3489 V 1813 3491 V 1809 3494 V 1805 3497 V 1800 3500 V 1796 3503 V 1792 3506 V 1788 3508 V 1783 3511 V 1779 3514 V 1775 3517 V 1771 3520 V 1766 3523 V 1762 3525 V 1758 3528 V 1754 3531 V 1749 3534 V 1745 3537 V 1741 3540 V 1737 3542 V 1732 3545 V 1728 3548 V 1724 3551 V 1720 3554 V 1715 3557 V 1711 3559 V 1707 3562 V 1703 3565 V 1698 3568 V 1694 3571 V 1690 3574 V 1689 3571 7 7 v 1687 3565 V 1684 3558 V 1681 3552 V 1678 3546 V 1675 3539 V 1672 3533 V 1670 3527 V 1667 3520 V 1664 3514 V 1661 3507 V 1658 3501 V 1655 3495 V 1653 3488 V 1650 3482 V 1647 3476 V 1644 3469 V 1641 3463 V 1638 3456 V 1636 3450 V 1633 3444 V 1630 3437 V 1627 3431 V 1624 3424 V 1621 3418 V 1619 3412 V 1619 3406 V 1616 3399 V 1613 3393 V 1610 3387 V 1607 3380 V 1604 3374 V 1602 3368 V 1599 3361 V 1596 3355 V 1593 3348 V 1590 3342 V 1587 3336 V 1585 3329 V 1582 3323 V 1579 3317 V 1576 3310 V 1573 3304 V 1570 3297 V 1568 3291 V 1565 3285 V 1562 3278 V 1559 3272 V 1556 3266 V 1553 3259 V 1551 3253 V 1548 3246 V 1548 3240 7 5 v 1545 3236 V 1542 3232 V 1539 3228 V 1536 3223 V 1534 3219 V 1531 3215 V 1528 3211 V 1525 3206 V 1522 3202 V 1519 3198 V 1517 3194 V 1514 3189 V 1511 3185 V 1508 3181 V 1505 3177 V 1502 3172 V 1500 3168 V 1497 3164 V 1494 3160 V 1491 3155 V 1488 3151 V 1485 3147 V 1483 3143 V 1480 3138 V 1477 3134 V 1474 3130 V 1471 3126 V 1468 3121 V 1465 3117 V 1463 3113 V 1460 3109 V 1457 3104 V 1454 3100 V 1646 2815 a(j)2212 3052 y(j)1976 3618 y(j)1409 3382 y(j)p 1619 3099 7 71 v 1619 3099 V 1622 3031 71 7 v 1784 3028 7 71 v 1784 3028 V 1787 2960 71 7 v 1929 3031 V 1997 3099 7 71 v 2000 3197 71 7 v 2067 3264 7 71 v 1997 3406 V 1929 3409 71 7 v 1831 3477 7 71 v 1764 3480 71 7 v 1622 3409 V 1619 3406 7 71 v 1551 3244 71 7 v 1548 3240 7 71 v 1716 2829 a Fh(1,0)159 b(1,1)1334 3065 y(1,-1)2189 3154 y(0,1)1334 3302 y(0,-1)2189 3391 y(-1,1)1806 3627 y(-1,0)-356 b(-1,-1)1898 3084 y(1,2)1948 3249 y(-1,2)1877 3373 y(-2,1)1691 3443 y(-2,-1)1646 3373 y(-1,-2)1575 3207 y(1,-2)1646 3084 y(2,-1)1811 3013 y(2,1)1081 3928 y Fw(A)33 b(connected)h(comp)s(onen)m (t)f(of)f Fs(H)2333 3943 y Fr(1)2372 3928 y Fw(.)-360 4163 y Fu(De\014nition)k(2.4.2)49 b Fw(Let)33 b Fs(G)28 b Fw(=)f(\()p Fs(V)5 b(;)17 b(E)6 b Fw(\))33 b(b)s(e)f(a)h(lo)s(cally)d (\014nite)i(undirected)h(graph.)44 b(Let)551 4404 y Fs(r)s Fw(\()p Fs(G)p Fw(\))27 b(=)h(sup)1045 4323 y Fm(\010)1113 4309 y(X)1049 4525 y Fo(f)p Fp(u;v)r Fo(g2)p Fp(E)1336 4404 y Fs(f)11 b Fw(\()p Fs(u)p Fw(\))p Fs(f)g Fw(\()p Fs(v)t Fw(\);)1757 4309 y Fm(X)1757 4520 y Fp(v)r Fo(2)p Fp(V)1916 4404 y Fs(f)g Fw(\()p Fs(v)t Fw(\))2102 4362 y Fr(2)2168 4404 y Fw(=)28 b(1)p Fs(;)49 b(f)38 b Fw(:)28 b Fs(V)49 b Ft(!)28 b Fq(R)2837 4323 y Fm(\011)2902 4404 y Fs(:)-360 4738 y Fw(The)34 b(follo)m(wing)29 b(Theorem)k(giv)m(es)g (us)g(an)g(upp)s(er)g(b)s(ound)g(for)f Fs(r)s Fw(\()p Fs(G)p Fw(\))g(of)g(a)g(graph)h Fs(G)p Fw(.)-360 4964 y Fu(Theorem)k(2.4.3)49 b Fj(L)-5 b(et)44 b Fs(\025)f Fj(b)-5 b(e)43 b(a)h(lab)-5 b(eling)42 b(of)h(ar)-5 b(cs)43 b(of)g Fs(G)g Fj(with)g(p)-5 b(ositive)43 b(r)-5 b(e)g(al)43 b(numb)-5 b(ers)43 b(such)g(that)h Fs(\025)p Fw(\()p Fs(u;)17 b(v)t Fw(\))42 b(=)-360 5084 y(1)p Fs(=\025)p Fw(\()p Fs(v)t(;)17 b(u)p Fw(\))p Fj(.)43 b(Then)1165 5252 y Fs(r)s Fw(\()p Fs(G)p Fw(\))27 b Ft(\024)1507 5185 y Fw(1)p 1507 5229 49 4 v 1507 5321 a(2)1583 5252 y(sup)1584 5334 y Fp(u)p Fo(2)p Fp(V)1795 5158 y Fm(X)1746 5373 y Fp(v)r Fo(2)p Fp(N)7 b Fr(\()p Fp(u)p Fr(\))2005 5252 y Fs(\025)p Fw(\()p Fs(u;)17 b(v)t Fw(\))p Fs(:)p eop %%Page: 29 33 29 32 bop -180 68 a Fk(2.4.)76 b(THE)33 b(RA)-8 b(YLEIGH)33 b(QUOTIENT)h(OF)e(OPERA)-8 b(TORS)1715 b Fw(29)-180 400 y Fu(Pro)s(of)37 b(:)-180 520 y Fw(Using)32 b(the)h(inequalit)m(y)f(b)s (et)m(w)m(een)i(arithmetic)d(and)i(geometric)e(mean)h(and)h(the)g(prop) s(ert)m(y)g(of)f Fs(\025)h Fw(w)m(e)g(obtain)642 686 y Fm(X)579 902 y Fo(f)p Fp(u;v)r Fo(g2)p Fp(E)866 781 y Fs(f)11 b Fw(\()p Fs(u)p Fw(\))p Fs(f)g Fw(\()p Fs(v)t Fw(\))26 b Ft(\024)1437 686 y Fm(X)1374 902 y Fo(f)p Fp(u;v)r Fo(g2)p Fp(E)1671 713 y Fw(1)p 1671 758 49 4 v 1671 849 a(2)1746 700 y Fm(\000)1792 781 y Fs(f)11 b Fw(\()p Fs(u)p Fw(\))1983 740 y Fr(2)2021 781 y Fs(\025)p Fw(\()p Fs(u;)17 b(v)t Fw(\))k(+)h Fs(f)11 b Fw(\()p Fs(v)t Fw(\))2610 740 y Fr(2)2649 781 y Fs(\025)p Fw(\()p Fs(v)t(;)17 b(u)p Fw(\))2933 700 y Fm(\001)3005 781 y Fw(=)27 b(\()p Ft(\003)p Fw(\))p Fs(:)-180 1106 y Fw(No)m(w)508 1321 y(\()p Ft(\003)p Fw(\))g(=)775 1253 y(1)p 775 1298 V 775 1389 a(2)851 1226 y Fm(X)850 1438 y Fp(u)p Fo(2)p Fp(V)1011 1321 y Fs(f)11 b Fw(\()p Fs(u)p Fw(\))1202 1280 y Fr(2)1307 1226 y Fm(X)1258 1442 y Fp(v)r Fo(2)p Fp(N)c Fr(\()p Fp(u)p Fr(\))1517 1321 y Fs(\025)p Fw(\()p Fs(u;)17 b(v)t Fw(\))26 b Ft(\024)1942 1253 y Fw(1)p 1942 1298 V 1942 1389 a(2)2018 1120 y Fm(0)2018 1300 y(@)2105 1321 y Fw(sup)2106 1402 y Fp(u)p Fo(2)p Fp(V)2318 1226 y Fm(X)2268 1442 y Fp(v)r Fo(2)p Fp(N)7 b Fr(\()p Fp(u)p Fr(\))2528 1321 y Fs(\025)p Fw(\()p Fs(u;)17 b(v)t Fw(\))2812 1120 y Fm(1)2812 1300 y(A)2914 1226 y(X)2914 1438 y Fp(u)p Fo(2)p Fp(V)3075 1321 y Fs(f)11 b Fw(\()p Fs(u)p Fw(\))3266 1280 y Fr(2)3305 1321 y Fs(:)3943 1607 y Fi(\004)-180 1799 y Fw(No)m(w)33 b(w)m(e)h(use)f(Theorem)g(2.4.3)f (to)g(compute)h Fs(r)s Fw(\()p Fs(H)1680 1814 y Fr(1)1719 1799 y Fw(\))-180 2022 y Fu(Theorem)k(2.4.4)1519 2145 y Fs(r)s Fw(\()p Fs(H)1685 2160 y Fr(1)1724 2145 y Fw(\))28 b Ft(\024)g Fw(1)p Fs(=)p Fw(2)22 b(+)2162 2057 y Ft(p)p 2245 2057 V 88 x Fw(2)o Fs(:)-180 2334 y Fu(Pro)s(of)37 b(:)-180 2454 y Fw(F)-8 b(rom)38 b(Euclid's)i(algorithm)d(it)i(follo)m (ws)f(that)i(for)f(an)m(y)i Fs(d)f Ft(2)g Fq(N)55 b Fw(the)40 b(v)m(ertices)h(\()p Fs(a;)17 b(b)p Fw(\))41 b(suc)m(h)g(that)f(gcd\()p Fs(a;)17 b(b)p Fw(\))40 b(=)g Fs(d)-180 2575 y Fw(form)c(a)h(connected) i(comp)s(onen)m(t)f(of)f Fs(H)1302 2590 y Fr(1)1341 2575 y Fw(.)58 b(These)39 b(connected)h(comp)s(onen)m(ts)e(are)f(isomorphic) f(and)h(hence)i(it)d(is)-180 2695 y(enough)d(to)g(consider)g(only)g (one)g(of)f(them,)h(e.g.)45 b(the)34 b(one)f(with)f Fs(d)c Fw(=)h(1.)44 b(The)34 b(mappings)e(\()p Fs(x;)17 b(y)t Fw(\))27 b Ft(!)h Fw(\()p Ft(\000)p Fs(y)t(;)17 b(x)p Fw(\))33 b(and)-180 2815 y(\()p Fs(x;)17 b(y)t Fw(\))27 b Ft(!)g Fw(\()p Fs(y)t(;)17 b(x)p Fw(\))32 b(are)h(automorphisms)e(of) h Fs(H)1499 2830 y Fr(1)1538 2815 y Fw(.)44 b(Consider)33 b(the)g(follo)m(wing)d(lab)s(eling)g Fs(\025)i Fw(of)g(arcs)i(\(the)f (arcs)g(in)f(the)-180 2936 y(same)g(orbits)g(ha)m(v)m(e)i(the)f(same)g (lab)s(els\))-35 3142 y Ft(\017)49 b Fw(for)32 b(the)h(self)f(lo)s(op)f Fs(e)i Fw(at)f(\(0)p Fs(;)17 b Fw(1\))32 b(let)g Fs(\025)p Fw(\()p Fs(e)p Fw(\))27 b(=)h(1)-35 3349 y Ft(\017)49 b Fw(for)32 b(the)h(arc)f Fs(e)h Fw(from)f(\(0)p Fs(;)17 b Fw(1\))31 b(to)h(\(1)p Fs(;)17 b Fw(1\))32 b(let)g Fs(\025)p Fw(\()p Fs(e)p Fw(\))c(=)f Fs(a)33 b Fw(\(the)g(opp)s(osite)f (arcs)h(ha)m(v)m(e)h(lab)s(el)d(1)p Fs(=a)p Fw(\))-35 3555 y Ft(\017)49 b Fw(for)32 b(an)g(arc)h Fs(e)g Fw(from)e Fs(u)h Fw(to)g Fs(v)37 b Fw(whic)m(h)c(w)m(as)g(not)g(lab)s(eled)e(y)m (et)i(let)1346 3890 y Fs(\025)p Fw(\()p Fs(e)p Fw(\))27 b(=)1655 3719 y Fm(\()1777 3769 y Fw(1)173 b(if)31 b Ft(jj)p Fs(u)p Ft(jj)2256 3784 y Fo(1)2357 3769 y Fw(=)c Ft(jj)p Fs(v)t Ft(jj)2623 3784 y Fo(1)1777 3889 y Fs(b)181 b Fw(if)31 b Ft(jj)p Fs(u)p Ft(jj)2256 3904 y Fo(1)2357 3889 y Fs(<)c Ft(jj)p Fs(v)t Ft(jj)2623 3904 y Fo(1)1777 4009 y Fw(1)p Fs(=b)83 b Fw(otherwise)-180 4273 y(The)40 b(lab)s(eling)35 b(satis\014es)40 b(the)f(conditions)f(of)g(Theorem)h (2.4.3.)61 b(F)-8 b(or)38 b Fs(v)k Fw(=)37 b(\(0)p Fs(;)17 b Fw(1\))38 b(w)m(e)i(ha)m(v)m(e)3255 4198 y Fm(P)3360 4302 y Fp(u)p Fo(2)p Fp(N)7 b Fr(\()p Fp(v)r Fr(\))3623 4273 y Fs(\025)p Fw(\()p Fs(v)t(;)17 b(u)p Fw(\))37 b(=)-180 4402 y(2)p Fs(a)27 b Fw(+)g(2,)42 b(for)d Fs(v)44 b Fw(=)c(\(1)p Fs(;)17 b Fw(1\))40 b(w)m(e)h(ha)m(v)m(e)1172 4327 y Fm(P)1277 4431 y Fp(u)p Fo(2)p Fp(N)7 b Fr(\()p Fp(v)r Fr(\))1540 4402 y Fs(\025)p Fw(\()p Fs(v)t(;)17 b(u)p Fw(\))39 b(=)i(2)p Fs(=a)26 b Fw(+)i(2)p Fs(b)p Fw(.)65 b(F)-8 b(or)40 b(the)g(other)g(v)m(ertices)h(t)m(w)m(o)g(neigh)m(b)s (ors)-180 4530 y(ha)m(v)m(e)35 b(larger,)e(one)h(has)h(smaller)d(and)i (one)g(has)g(equal)g(in\014nit)m(y)f(norm)g(\(for)g(v)m(ertex)j(\()p Fs(x;)17 b(y)t Fw(\),)33 b(w.l.o.g.)47 b Fs(x)31 b(>)e(y)k(>)d Fw(0,)-180 4651 y(then)h(\()p Fs(x)19 b Fw(+)g Fs(y)t(;)e(y)t Fw(\))p Fs(;)g Fw(\()p Fs(x;)g(x)g Fw(+)h Fs(y)t Fw(\))30 b(ha)m(v)m(e)i(larger,)e(\()p Fs(x;)17 b(y)22 b Ft(\000)d Fs(x)p Fw(\))32 b(has)f(equal)g(and)g(\()p Fs(x)19 b Ft(\000)g Fs(y)t(;)e(y)t Fw(\))28 b(has)k(smaller)c(in\014nit)m(y)j (norm\).)-180 4771 y(Hence)109 4696 y Fm(P)214 4800 y Fp(u)p Fo(2)p Fp(N)7 b Fr(\()p Fp(v)r Fr(\))478 4771 y Fs(\025)p Fw(\()p Fs(v)t(;)17 b(u)p Fw(\))26 b(=)i(1)20 b(+)g(2)p Fs(b)g Fw(+)h(1)p Fs(=b)p Fw(.)43 b(F)-8 b(or)31 b Fs(b)d Fw(=)f(1)p Fs(=)1917 4689 y Ft(p)p 2000 4689 V 82 x Fw(2)k(and)h Fs(a)c Fw(=)2452 4689 y Ft(p)p 2535 4689 V 82 x Fw(2)20 b Ft(\000)h Fw(1)p Fs(=)p Fw(2)31 b(w)m(e)h(ha)m(v)m(e)h(max\(2)p Fs(a)20 b Fw(+)h(2)p Fs(;)c Fw(2)p Fs(=a)i Fw(+)-180 4915 y(2)p Fs(b;)e Fw(1)22 b(+)g(2)p Fs(b)g Fw(+)g(1)p Fs(=b)p Fw(\))28 b(=)f(1)22 b(+)g(2)859 4833 y Ft(p)p 942 4833 V 82 x Fw(2)32 b(and)h(hence)h Fs(r)s Fw(\()p Fs(H)1650 4930 y Fr(1)1689 4915 y Fw(\))28 b Ft(\024)g Fw(1)p Fs(=)p Fw(2)21 b(+)2126 4833 y Ft(p)p 2209 4833 V 82 x Fw(2.)1658 b Fi(\004)-80 5178 y Fw(The)33 b(op)s(erator)f Fs(T)584 5142 y Fo(0)640 5178 y Fw(men)m(tioned)g(at)g (the)h(end)g(of)f(the)h(pro)s(of)f(of)g(Theorem)h(2.3.3)f(w)m(as)i (de\014ned)g(as)1045 5400 y(\()p Fs(T)1154 5359 y Fo(0)1177 5400 y Fs(f)11 b Fw(\)\()p Fs(x;)17 b(y)t Fw(\))26 b(=)i Fs(f)11 b Fw(\()p Fs(x)22 b Ft(\000)h Fw(2)p Fs(y)t(;)17 b(y)t Fw(\))i(+)j Fs(f)11 b Fw(\()p Fs(x;)17 b(y)25 b Ft(\000)e Fw(2)p Fs(x)p Fw(\))p Fs(:)p eop %%Page: 30 34 30 33 bop -360 68 a Fw(30)57 b Fk(CHAPTER)34 b(2.)76 b(HARMONIC)33 b(ANAL)-8 b(YSIS)34 b(O)m(VER)f(LOCALL)-8 b(Y)33 b(COMP)-8 b(A)m(CT)34 b(ABELIAN)g(GR)m(OUPS)-360 400 y Fw(Hence)g(b)m(y)f(Lemma)f(2.4.1)866 520 y(\()924 495 y Fm(b)904 520 y Fs(T)975 491 y Fo(0)1016 494 y Fm(b)998 520 y Fs(f)11 b Fw(\)\()p Fs(x;)17 b(y)t Fw(\))27 b(=)1470 494 y Fm(b)1452 520 y Fs(f)11 b Fw(\()p Fs(x;)17 b(x)23 b Fw(+)f(2)p Fs(y)t Fw(\))f(+)2100 494 y Fm(b)2082 520 y Fs(f)10 b Fw(\()p Fs(x)23 b Fw(+)f(2)p Fs(y)t(;)17 b(y)t Fw(\))p Fs(:)-360 690 y Fw(Again)33 b(w)m(e)j(can)e(sho)m(w)i(an) e(upp)s(er)h(b)s(ound)f(on)g(the)h(Ra)m(yleigh)e(quotien)m(t)i(of)e Fs(T)2497 654 y Fo(0)2555 690 y Fw(in)g(the)i(space)g Fs(X)42 b Fw(b)m(y)36 b(considering)-360 810 y(graph)c Fs(H)-3 825 y Fr(2)69 810 y Fw(for)f(whic)m(h)i Fs(r)s Fw(\()p Fs(H)662 825 y Fr(2)701 810 y Fw(\))28 b(=)f Fs(r)s Fw(\()971 785 y Fm(b)955 810 y Fs(T)1026 774 y Fo(0)1049 810 y Fw(\))h(=)f Fs(r)s Fw(\()p Fs(T)1374 774 y Fo(0)1397 810 y Fw(\).)43 b(Let)33 b Fs(H)1761 825 y Fr(2)1832 810 y Fw(b)s(e)g(the)f(undirected)h(graph)g(with)e(v)m (ertices)j Fq(Z)3534 774 y Fr(2)3592 810 y Ft(\000)23 b(f)p Fw(0)p Ft(g)-360 931 y Fw(and)33 b(\()p Fs(a;)17 b(b)p Fw(\))32 b(connected)j(with)d(\()p Fs(a)22 b Fw(+)g(2)p Fs(b;)17 b(b)p Fw(\))33 b(and)g(\()p Fs(a;)17 b(b)22 b Fw(+)g(2)p Fs(a)p Fw(\).)1445 1485 y Fa(u)186 b(u)1445 1249 y(u)g(u)p 1445 1252 237 7 v 1441 1485 7 237 v 1445 1489 237 7 v 1678 1485 7 237 v 1681 1131 a(u)1799 1249 y(u)1799 1485 y(u)1681 1603 y(u)-286 b(u)1327 1485 y(u)1327 1249 y(u)1445 1131 y(u)p 1678 1249 7 119 v 1681 1252 119 7 v 1681 1489 V 1678 1603 7 119 v 1441 1603 V 1327 1489 119 7 v 1327 1252 V 1441 1249 7 119 v 1678 1131 7 71 v 1681 1134 71 7 v 1678 1131 7 3 v 1680 1128 V 1683 1125 V 1686 1122 V 1689 1120 V 1692 1117 V 1695 1114 V 1697 1111 V 1700 1108 V 1703 1105 V 1706 1103 V 1709 1100 V 1712 1097 V 1714 1094 V 1717 1091 V 1720 1088 V 1723 1086 V 1799 1252 71 7 v 1796 1249 7 71 v 1796 1249 7 3 v 1799 1246 V 1801 1243 V 1804 1241 V 1807 1238 V 1810 1235 V 1813 1232 V 1816 1229 V 1818 1226 V 1821 1224 V 1824 1221 V 1827 1218 V 1830 1215 V 1833 1212 V 1835 1209 V 1838 1207 V 1841 1204 V 1799 1489 71 7 v 1796 1556 7 71 v 1796 1488 7 3 v 1799 1491 V 1801 1494 V 1804 1497 V 1807 1499 V 1810 1502 V 1813 1505 V 1816 1508 V 1818 1511 V 1821 1514 V 1824 1516 V 1827 1519 V 1830 1522 V 1833 1525 V 1835 1528 V 1838 1531 V 1841 1533 V 1678 1674 7 71 v 1681 1607 71 7 v 1678 1606 7 3 v 1680 1609 V 1683 1612 V 1686 1615 V 1689 1618 V 1692 1620 V 1695 1623 V 1697 1626 V 1700 1629 V 1703 1632 V 1706 1635 V 1709 1637 V 1712 1640 V 1714 1643 V 1717 1646 V 1720 1649 V 1723 1652 V 1441 1674 7 71 v 1374 1607 71 7 v 1441 1606 7 3 v 1439 1609 V 1436 1612 V 1433 1615 V 1430 1618 V 1427 1620 V 1424 1623 V 1422 1626 V 1419 1629 V 1416 1632 V 1413 1635 V 1410 1637 V 1407 1640 V 1405 1643 V 1402 1646 V 1399 1649 V 1396 1652 V 1256 1489 71 7 v 1323 1556 7 71 v 1323 1488 7 3 v 1320 1491 V 1318 1494 V 1315 1497 V 1312 1499 V 1309 1502 V 1306 1505 V 1303 1508 V 1301 1511 V 1298 1514 V 1295 1516 V 1292 1519 V 1289 1522 V 1286 1525 V 1284 1528 V 1281 1531 V 1278 1533 V 1256 1252 71 7 v 1323 1249 7 71 v 1323 1249 7 3 v 1320 1246 V 1318 1243 V 1315 1241 V 1312 1238 V 1309 1235 V 1306 1232 V 1303 1229 V 1301 1226 V 1298 1224 V 1295 1221 V 1292 1218 V 1289 1215 V 1286 1212 V 1284 1209 V 1281 1207 V 1278 1204 V 1441 1131 7 71 v 1374 1134 71 7 v 1441 1131 7 3 v 1439 1128 V 1436 1125 V 1433 1122 V 1430 1120 V 1427 1117 V 1424 1114 V 1422 1111 V 1419 1108 V 1416 1105 V 1413 1103 V 1410 1100 V 1407 1097 V 1405 1094 V 1402 1091 V 1399 1088 V 1396 1086 V 1558 1216 a Fh(1,-1)1468 1541 y(-1,1)1225 1305 y(3,1)520 b(1,-3)1558 1098 y(3,-1)1823 1452 y(-1,-3)1468 1659 y(-1,3)1204 1452 y(-3,1)1468 1187 y(3,1)1468 1305 y(1,1)1537 1452 y(-1,-1)1537 1594 y(-3,-1)2153 1249 y Fa(u)2035 1367 y(u)186 b(u)p 2032 1370 V 2029 1373 V 2026 1376 V 2023 1379 V 2021 1381 V 2018 1384 V 2015 1387 V 2012 1390 V 2009 1393 V 2006 1396 V 2004 1398 V 2001 1401 V 1998 1404 V 1995 1407 V 1992 1410 V 1989 1413 V 1987 1415 V 2032 1438 7 71 v 2032 1370 7 3 v 2035 1373 V 2038 1376 V 2040 1379 V 2043 1381 V 2046 1384 V 2049 1387 V 2052 1390 V 2055 1393 V 2057 1396 V 2060 1398 V 2063 1401 V 2066 1404 V 2069 1407 V 2072 1410 V 2074 1413 V 2077 1415 V 2268 1370 V 2265 1373 V 2262 1376 V 2260 1379 V 2257 1381 V 2254 1384 V 2251 1387 V 2248 1390 V 2245 1393 V 2243 1396 V 2240 1398 V 2237 1401 V 2234 1404 V 2231 1407 V 2228 1410 V 2226 1413 V 2223 1415 V 2268 1438 7 71 v 2268 1370 7 3 v 2271 1373 V 2274 1376 V 2277 1379 V 2280 1381 V 2282 1384 V 2285 1387 V 2288 1390 V 2291 1393 V 2294 1396 V 2297 1398 V 2299 1401 V 2302 1404 V 2305 1407 V 2308 1410 V 2311 1413 V 2314 1415 V 2150 1252 V 2147 1255 V 2144 1258 V 2142 1260 V 2139 1263 V 2136 1266 V 2133 1269 V 2130 1272 V 2127 1275 V 2125 1277 V 2122 1280 V 2119 1283 V 2116 1286 V 2113 1289 V 2110 1292 V 2108 1294 V 2105 1297 V 2102 1300 V 2099 1303 V 2096 1306 V 2093 1309 V 2091 1311 V 2088 1314 V 2085 1317 V 2082 1320 V 2079 1323 V 2076 1326 V 2074 1328 V 2071 1331 V 2068 1334 V 2065 1337 V 2062 1340 V 2059 1343 V 2057 1345 V 2054 1348 V 2051 1351 V 2048 1354 V 2045 1357 V 2042 1360 V 2040 1362 V 2037 1365 V 2034 1368 V 2150 1252 V 2153 1255 V 2156 1258 V 2159 1260 V 2161 1263 V 2164 1266 V 2167 1269 V 2170 1272 V 2173 1275 V 2176 1277 V 2178 1280 V 2181 1283 V 2184 1286 V 2187 1289 V 2190 1292 V 2193 1294 V 2195 1297 V 2198 1300 V 2201 1303 V 2204 1306 V 2207 1309 V 2210 1311 V 2212 1314 V 2215 1317 V 2218 1320 V 2221 1323 V 2224 1326 V 2227 1328 V 2229 1331 V 2232 1334 V 2235 1337 V 2238 1340 V 2241 1343 V 2244 1345 V 2246 1348 V 2249 1351 V 2252 1354 V 2255 1357 V 2258 1360 V 2261 1362 V 2263 1365 V 2266 1368 V 2114 1216 a Fh(1,0)2232 1334 y(1,2)-324 b(1,-2)2153 1202 y Fa(j)1314 1866 y Fw(A)33 b(part)f(of)g(graph)h Fs(H)2100 1881 y Fr(2)2139 1866 y Fw(.)-360 2082 y Fu(Theorem)k(2.4.5) 1473 2202 y Fs(r)s Fw(\()p Fs(H)1639 2217 y Fr(2)1678 2202 y Fw(\))27 b Ft(\024)1848 2115 y(p)p 1931 2115 49 4 v 87 x Fw(3)p Fs(:)-360 2387 y Fu(Pro)s(of)37 b(:)-360 2507 y Fw(The)j(mappings)d(\()p Fs(x;)17 b(y)t Fw(\))37 b Ft(!)h Fw(\()p Ft(\000)p Fs(y)t(;)17 b(x)p Fw(\))38 b(and)h(\()p Fs(x;)17 b(y)t Fw(\))37 b Ft(!)g Fw(\()p Fs(y)t(;)17 b(x)p Fw(\))38 b(are)h(automorphisms)e(of)h Fs(H)2955 2522 y Fr(1)2994 2507 y Fw(.)62 b(Ev)m(ery)40 b(v)m(ertex)h Fs(u)c Fw(=)-360 2628 y(\()p Fs(x;)17 b(y)t Fw(\))28 b(where)j(b)s(oth)e Fs(x;)17 b(y)33 b Fw(are)c(non-zero,)h Ft(j)p Fs(x)p Ft(j)d(6)p Fw(=)h Ft(j)p Fs(y)t Ft(j)g Fw(is)h(connected)i(to)e(three)h(v)m(ertices)g(with)f(larger)f(and)i (one)f(with)-360 2748 y(smaller)d Ft(1)h Fw(norm.)42 b(If)27 b(one)i(of)e(the)h Fs(x;)17 b(y)32 b Fw(is)27 b(zero)h(or)g Ft(j)p Fs(x)p Ft(j)f Fw(=)h Ft(j)p Fs(y)t Ft(j)e Fw(then)j Fs(u)e Fw(is)h(connected)h(to)f(t)m(w)m(o)g(v)m (ertices)i(with)d(larger)-360 2869 y(and)j(t)m(w)m(o)g(v)m(ertices)g (with)f(equal)h Ft(1)f Fw(norm)f(\(p)s(ossibly)h(via)g(a)g(self-lo)s (op\).)40 b(Lab)s(el)28 b(the)i(arc)g Fs(e)f Fw(b)s(et)m(w)m(een)j(the) e(v)m(ertices)-360 2989 y Fs(u;)17 b(v)1039 3222 y(\025)p Fw(\()p Fs(e)p Fw(\))28 b(=)1348 3052 y Fm(\()1470 3101 y Fw(1)183 b(if)31 b Ft(jj)p Fs(u)p Ft(jj)1959 3116 y Fo(1)2060 3101 y Fw(=)c Ft(jj)p Fs(v)t Ft(jj)2326 3116 y Fo(1)1470 3221 y Fs(a)181 b Fw(if)31 b Ft(jj)p Fs(u)p Ft(jj)1959 3236 y Fo(1)2060 3221 y Fs(<)c Ft(jj)p Fs(v)t Ft(jj)2326 3236 y Fo(1)1470 3342 y Fw(1)p Fs(=a)83 b Fw(otherwise)-360 3526 y(F)-8 b(or)34 b(an)m(y)h Fs(u)f Fw(w)m(e)h(ha)m(v)m(e)h(either)744 3451 y Fm(P)849 3555 y Fp(v)r Fo(2)p Fp(N)7 b Fr(\()p Fp(u)p Fr(\))1112 3526 y Fs(\025)p Fw(\()p Fs(u;)17 b(v)t Fw(\))30 b(=)h(3)p Fs(a)23 b Fw(+)h(1)p Fs(=a)34 b Fw(or)2060 3451 y Fm(P)2165 3555 y Fp(v)r Fo(2)p Fp(N)7 b Fr(\()p Fp(u)p Fr(\))2428 3526 y Fs(\025)p Fw(\()p Fs(u;)17 b(v)t Fw(\))30 b(=)h(2)23 b(+)h(2)p Fs(a)p Fw(.)49 b(F)-8 b(or)34 b Fs(a)d Fw(=)3563 3444 y Ft(p)p 3646 3444 V 82 x Fw(3)j(w)m(e)-360 3670 y(obtain)d Fs(r)s Fw(\()p Fs(H)109 3685 y Fr(2)148 3670 y Fw(\))d Ft(\024)319 3588 y(p)p 402 3588 V 82 x Fw(3.)3285 b Fi(\004)-260 3933 y Fw(It)39 b(is)g(not)g(hard)g(to)g(\014nd)g(lab)s (elings)e(of)i(the)g(v)m(ertices)i(of)e Fs(H)1960 3948 y Fr(1)2038 3933 y Fw(and)g Fs(H)2315 3948 y Fr(2)2394 3933 y Fw(whic)m(h)g(sho)m(w)i(that)e(the)g(constan)m(ts)i(in)-360 4053 y(Theorems)33 b(2.4.4)f(and)h(2.4.5)f(are)h(optimal.)-360 4384 y Fn(2.5)161 b(Lattice)54 b(Dualit)l(y:)72 b(Banaszczyk's)53 b(T)-13 b(ransference)52 b(Theorem)-360 4650 y Fw(Giv)m(en)32 b(an)h(an)f Fs(n)23 b Ft(\002)f Fs(n)33 b Fw(regular)f(matrix)f Fs(B)37 b Fw(a)c(lattice)e Fs(L)i Fw(is)1327 4860 y Fs(L)28 b Fw(=)f Ft(f)p Fs(B)5 b(x)17 b Fw(;)34 b Fs(x)28 b Ft(2)g Fq(Z)2032 4819 y Fp(n)2076 4860 y Ft(g)p Fs(:)-360 5070 y Fw(Alternativ)m(ely)35 b(a)g(lattice)f(can)i(b)s(e)g(view)m(ed)h(as)f (a)f(discrete)i(additiv)m(e)e(subgroup)h(of)f Fq(R)2812 5034 y Fp(n)2865 5070 y Fw(.)53 b(De\014ne)36 b(the)g(successiv)m(e) -360 5190 y(minima)29 b Fs(\025)49 5205 y Fr(1)89 5190 y Fs(;)17 b(:)g(:)g(:)e(;)i(\025)364 5205 y Fp(n)443 5190 y Fw(of)33 b(the)g(lattice)e Fs(L)871 5400 y(\025)928 5415 y Fp(i)984 5400 y Fw(=)c(min)o Ft(f)p Fs(r)j(>)e Fw(0;)17 b(dim)d(span)q(\()p Fs(L)23 b Ft(\\)f Fs(r)s(B)2281 5415 y Fp(n)2328 5400 y Fw(\))27 b Ft(\025)i Fs(i)p Ft(g)p Fs(;)p eop %%Page: 31 35 31 34 bop -180 69 a Fk(2.5.)76 b(LA)-8 b(TTICE)34 b(DUALITY:)f (BANASZCZYK'S)g(TRANSFERENCE)i(THEOREM)722 b Fw(31)-180 400 y(where)31 b Fs(B)173 415 y Fp(n)251 400 y Fw(is)e(the)i(unit)e (ball)f(in)i Fq(R)1080 363 y Fp(n)1133 400 y Fw(.)42 b(The)31 b(dual)f(lattice)e Fs(L)1976 363 y Fo(\003)2046 400 y Fw(of)i(the)h(lattice)d Fs(L)j Fw(is)e(the)i(lattice)d(with)i (matrix)f Fs(B)3883 363 y Fo(\000)p Fp(T)3993 400 y Fw(.)-180 520 y(Our)k(goal)d(is)j(to)f(pro)m(v)m(e)i(the)f(T)-8 b(ransference)34 b(Theorem)f(of)f(Banaszczyk)j([Ban93])p -14 648 3869 4 v -14 1012 4 364 v 46 743 a Fu(Theorem)j(2.5.1)49 b Fj(F)-7 b(or)34 b(any)g(lattic)-5 b(e)35 b Fs(L)h Fj(in)e Fq(R)1745 706 y Fp(n)1486 963 y Fs(\025)1543 978 y Fp(i)1571 963 y Fw(\()p Fs(L)p Fw(\))p Fs(\025)1770 978 y Fp(n)p Fr(+1)p Fo(\000)p Fp(i)1986 963 y Fw(\()p Fs(L)2090 921 y Fo(\003)2130 963 y Fw(\))28 b Ft(\024)g Fs(n:)p 3851 1012 V -14 1015 3869 4 v -80 1185 a Fw(The)42 b(Theorem)g(is)f(tigh)m (t)f(up)i(to)f(a)g(m)m(ultiplicativ)m(e)d(constan)m(t)k(as)g(there)g (exist)g(a)f(self)g(dual)g(lattice)f Fs(L)h Fw(suc)m(h)-180 1305 y(that)e Fs(\025)95 1320 y Fr(1)135 1305 y Fw(\()p Fs(L)p Fw(\))277 1269 y Fr(2)356 1305 y Ft(\025)517 1266 y Fp(n)p 483 1283 111 4 v 483 1340 a Fr(2)p Fp(\031)r(e)603 1305 y Fw(\(1)27 b(+)g Fs(o)p Fw(\(1\)\))39 b(as)g Fs(n)h Ft(!)f(1)p Fw(,)i(a)e(result)g(of)g(Con)m(w)m(a)m(y)j(and)d(Thompson)h (\(see)h([Mil73)n(],)g(p.)64 b(42\).)-180 1426 y(A)38 b(transference)i(theorem)e(w)m(as)h(used)g(in)f([LLS90])g(to)g(sho)m(w) h(that)f Fs(O)s Fw(\()p Fs(n)p Fw(\)-appro)m(ximation)d(of)i(shortest)i (lattice)-180 1546 y(v)m(ector)34 b(in)d Fs(L)292 1510 y Fr(2)365 1546 y Fw(norm)g(cannot)i(b)s(e)g(NP-hard)g(unless)g (NP=co-NP)-8 b(.)-180 1714 y(In)33 b(addition)e(to)h(the)h(material)d (co)m(v)m(ered)k(in)e(section)h(2.1)f(w)m(e)i(will)c(only)i(need)h (that)g(c)m(haracters)h(of)e Fq(R)3567 1678 y Fp(n)3652 1714 y Fw(are)1668 1920 y Ft(f)p Fs(\037)1779 1935 y Fp(b)1813 1920 y Ft(j)p Fs(b)c Ft(2)g Fq(R)2070 1879 y Fp(n)2123 1920 y Ft(g)-180 2126 y Fw(where)34 b Fs(\037)163 2141 y Fp(b)197 2126 y Fw(\()p Fs(x)p Fw(\))28 b(=)g(exp)q(\()p Fs(b)688 2090 y Fp(T)743 2126 y Fs(x)p Fw(\))p Ft(g)p Fw(.)44 b(The)33 b(F)-8 b(ourier)32 b(transform)f(of)i(a)f(\014nite)g (measure)h Fs(\026)f Fw(is)g(de\014ned)i(as)1382 2380 y Fm(b)-59 b Fs(\026)p Fw(\()p Fs(x)p Fw(\))28 b(=)1699 2245 y Fm(Z)1754 2470 y Fp(y)r Fo(2)p Fc(R)1886 2451 y Fg(n)1950 2380 y Fs(\037)2011 2395 y Fp(x)2055 2380 y Fw(\()p Fs(y)t Fw(\))17 b Fs(d\026)p Fw(\()p Fs(y)t Fw(\))p Fs(:)-80 2717 y Fw(F)-8 b(or)31 b Fs(A)d Ft(\022)g Fq(R)366 2681 y Fp(n)452 2717 y Fw(let)1511 2923 y Fs(\032)p Fw(\()p Fs(A)p Fw(\))g(=)1842 2828 y Fm(X)1844 3040 y Fp(x)p Fo(2)p Fp(A)2002 2923 y Fs(e)2047 2882 y Fo(\000)p Fp(\031)r Fo(jj)p Fp(x)p Fo(jj)2265 2859 y Fh(2)2302 2923 y Fs(:)1490 b Fw(\(2.8\))-180 3217 y(Later)32 b(in)g(section)h (2.6)f(w)m(e)i(will)c(pro)m(v)m(e)k(the)f(follo)m(wing)d(Lemma.)-180 3431 y Fu(Lemma)37 b(2.5.2)49 b Fj(L)-5 b(et)44 b Fs(n)h Ft(\025)g Fw(2)p Fj(.)71 b(L)-5 b(et)44 b Fs(A)g Fj(b)-5 b(e)43 b(a)h(b)-5 b(al)5 b(l)43 b(of)h(diameter)2302 3392 y Fr(3)p 2302 3408 36 4 v 2302 3466 a(4)2347 3359 y Ft(p)p 2430 3359 59 4 v 72 x Fs(n)g Fj(c)-5 b(enter)g(e)g(d)43 b(ar)-5 b(ound)44 b(the)g(origin)f(and)g(let)-180 3552 y Fs(u)27 b Ft(2)h Fq(R)63 3515 y Fp(n)116 3552 y Fj(.)45 b(Then)1429 3734 y Fs(\032)1479 3653 y Fm(\000)1525 3734 y Fw(\()p Fs(L)23 b Fw(+)f Fs(u)p Fw(\))f Ft(n)h Fs(A)2010 3653 y Fm(\001)p 1429 3785 627 4 v 1646 3876 a Fs(\032)p Fw(\()p Fs(L)p Fw(\))2094 3808 y Ft(\024)28 b Fw(0)p Fs(:)p Fw(285)-80 4064 y(Giv)m(en)k(a)g(lattice)f Fs(L)i Fw(w)m(e)h(de\014ne)g(a)e(discrete)h(measure)g(on)g Fq(R)2127 4028 y Fp(n)2212 4064 y Fw(b)m(y)1514 4320 y Fs(\033)1569 4335 y Fp(L)1622 4320 y Fw(\()p Fs(A)p Fw(\))28 b(=)1912 4252 y Fs(\032)p Fw(\()p Fs(A)23 b Ft(\\)f Fs(L)p Fw(\))p 1912 4297 377 4 v 2004 4388 a Fs(\032)p Fw(\()p Fs(L)p Fw(\))2299 4320 y Fs(:)-180 4575 y Fw(Let)1526 4725 y Fs(\036)1584 4740 y Fp(L)1636 4725 y Fw(\()p Fs(u)p Fw(\))27 b(=)1909 4658 y Fs(\032)p Fw(\()p Fs(L)22 b Fw(+)g Fs(u)p Fw(\))p 1909 4702 369 4 v 1996 4794 a Fs(\032)p Fw(\()p Fs(L)p Fw(\))2287 4725 y Fs(:)-180 4944 y Fw(Later)32 b(in)g(section)h(2.6)f(w)m(e)i(will)c(pro)m(v)m(e)-180 5158 y Fu(Lemma)37 b(2.5.3)49 b Fj(F)-7 b(or)34 b(a)h(lattic)-5 b(e)35 b Fs(L)g Fj(and)f(its)h(dual)g(lattic)-5 b(e)35 b Fs(L)2046 5122 y Fo(\003)1718 5364 y Fm(c)-104 b Fs(\033)1769 5379 y Fp(L)1849 5364 y Fw(=)28 b Fs(\036)2011 5379 y Fp(L)2059 5360 y Fe(\003)2099 5364 y Fs(:)p eop %%Page: 32 36 32 35 bop -360 68 a Fw(32)57 b Fk(CHAPTER)34 b(2.)76 b(HARMONIC)33 b(ANAL)-8 b(YSIS)34 b(O)m(VER)f(LOCALL)-8 b(Y)33 b(COMP)-8 b(A)m(CT)34 b(ABELIAN)g(GR)m(OUPS)-360 400 y Fu(Pro)s(of)j(of)h(Theorem)f(2.5.1)-360 520 y Fw(F)-8 b(or)41 b Fs(n)i Fw(=)g(1,)h Fs(\025)221 535 y Fr(1)260 520 y Fw(\()p Fs(L)p Fw(\))p Fs(\025)459 535 y Fr(1)499 520 y Fw(\()p Fs(L)603 484 y Fo(\003)643 520 y Fw(\))f(=)g(1.)70 b(Assume)43 b(that)e Fs(n)j Ft(\025)f Fw(2.)71 b(Supp)s(ose)42 b(that)g(there)g(is)g(a)f(lattice)f Fs(L)i Fw(suc)m(h)h(that)-360 640 y Fs(\025)-303 655 y Fp(i)-275 640 y Fw(\()p Fs(L)p Fw(\))p Fs(\025)-76 655 y Fp(n)p Fr(+1)p Fo(\000)p Fp(i)140 640 y Fw(\()p Fs(L)244 604 y Fo(\003)284 640 y Fw(\))36 b Fs(>)g(n)p Fw(.)59 b(W)-8 b(e)38 b(can)g(scale)g(the)g(lattice)e(so)h (that)h Fs(\025)2084 655 y Fp(i)2112 640 y Fw(\()p Fs(L)p Fw(\))e Fs(>)2412 601 y Fr(3)p 2412 617 36 4 v 2412 675 a(4)2458 569 y Ft(p)p 2541 569 59 4 v 71 x Fs(n)i Fw(and)f Fs(\025)2888 655 y Fp(n)p Fr(+1)p Fo(\000)p Fp(i)3104 640 y Fw(\()p Fs(L)3208 604 y Fo(\003)3248 640 y Fw(\))f Fs(>)3444 601 y Fr(3)p 3444 617 36 4 v 3444 675 a(4)3489 569 y Ft(p)p 3572 569 59 4 v 71 x Fs(n)26 b Fw(+)3768 601 y Fr(4)p 3768 617 36 4 v 3768 675 a(5)3813 640 y Fw(.)-360 761 y(Let)38 b Fs(A)h Fw(and)f Fs(A)200 725 y Fo(\003)278 761 y Fw(b)s(e)g(the)h(balls)e(of)h(diameters)1401 722 y Fr(3)p 1401 738 V 1401 795 a(4)1447 689 y Ft(p)p 1530 689 59 4 v 72 x Fs(n)g Fw(and)1831 722 y Fr(3)p 1831 738 36 4 v 1831 795 a(4)1877 689 y Ft(p)p 1960 689 59 4 v 72 x Fs(n)26 b Fw(+)2156 722 y Fr(4)p 2156 738 36 4 v 2156 795 a(5)2239 761 y Fw(cen)m(tered)41 b(around)d(the)h (origin.)58 b(W)-8 b(e)39 b(ha)m(v)m(e)-360 881 y(dim)15 b(span)q(\()p Fs(L)31 b Ft(\\)g Fs(A)p Fw(\))49 b Fs(<)g(i)c Fw(and)h(dim)15 b(span\()p Fs(L)1291 845 y Fo(\003)1362 881 y Ft(\\)31 b Fs(A)1532 845 y Fo(\003)1572 881 y Fw(\))49 b Fs(<)f(n)31 b Fw(+)g(1)f Ft(\000)i Fs(i)p Fw(.)81 b(Hence)47 b(there)e(is)g(a)g(v)m(ector)h Fs(u)f Fw(whic)m(h)g(is)-360 1002 y(p)s(erp)s(endicular)29 b(to)g(all)e(v)m(ectors)32 b(in)c Fs(L)16 b Ft(\\)g Fs(A)31 b Fw(and)e(all)f(v)m(ectors)j(in)e Fs(L)2034 965 y Fo(\003)2090 1002 y Ft(\\)16 b Fs(A)2245 965 y Fo(\003)2285 1002 y Fw(.)42 b(W)-8 b(e)30 b(can)g(c)m(hose)h Fs(u)e Fw(suc)m(h)i(that)e Ft(jj)p Fs(u)p Ft(jj)d Fw(=)3768 962 y Fr(4)p 3768 979 V 3768 1036 a(5)3813 1002 y Fw(.)-215 1184 y Ft(\017)49 b Fw(Since)33 b Fs(u)f Fw(is)g(p)s(erp)s(endicular)g (to)g(all)e(v)m(ectors)k(in)e Fs(L)23 b Ft(\\)f Fs(A)p Fw(,)-33 1416 y Fs(\036)25 1431 y Fp(L)73 1412 y Fe(\003)113 1416 y Fw(\()p Fs(u)p Fw(\))27 b(=)i Fm(b)-57 b Fs(\033)430 1431 y Fp(L)483 1416 y Fw(\()p Fs(u)p Fw(\))27 b(=)746 1321 y Fm(X)750 1533 y Fp(x)p Fo(2)p Fp(L)906 1416 y Fs(\033)961 1431 y Fp(L)1014 1416 y Fw(\()p Fs(x)p Fw(\))p Fs(e)1190 1375 y Fo(\000)p Fr(2)p Fp(\031)r(iu)1388 1351 y Fg(T)1437 1375 y Fp(x)1508 1416 y Fw(=)1657 1321 y Fm(X)1612 1533 y Fp(x)p Fo(2)p Fp(L)p Fo(\\)p Fp(A)1863 1416 y Fs(\033)1918 1431 y Fp(L)1971 1416 y Fw(\()p Fs(x)p Fw(\))22 b(+)2262 1321 y Fm(X)2222 1537 y Fp(x)p Fo(2)p Fp(L)p Fo(n)p Fp(A)2462 1416 y Fs(\033)2517 1431 y Fp(L)2569 1416 y Fw(\()p Fs(x)p Fw(\))p Fs(e)2745 1375 y Fo(\000)p Fr(2)p Fp(\031)r(iu)2943 1351 y Fg(T)2992 1375 y Fp(x)3064 1416 y Ft(\025)1333 1607 y Fm(X)1287 1818 y Fp(x)p Fo(2)p Fp(L)p Fo(\\)p Fp(A)1538 1701 y Fs(\033)1593 1716 y Fp(L)1646 1701 y Fw(\()p Fs(x)p Fw(\))h Ft(\000)1939 1607 y Fm(X)1899 1823 y Fp(x)p Fo(2)p Fp(L)p Fo(n)p Fp(A)2139 1701 y Fs(\033)2194 1716 y Fp(L)2246 1701 y Fw(\()p Fs(x)p Fw(\))28 b(=)g(1)22 b Ft(\000)g Fw(2)2784 1607 y Fm(X)2745 1823 y Fp(x)p Fo(2)p Fp(L)p Fo(n)p Fp(A)2984 1701 y Fs(\033)3039 1716 y Fp(L)3092 1701 y Fw(\()p Fs(x)p Fw(\))28 b Ft(\025)g Fw(0)p Fs(:)p Fw(43)p Fs(:)82 b Fw(\(2.9\))-215 2040 y Ft(\017)49 b Fw(W)-8 b(e)33 b(ha)m(v)m(e)850 2206 y Fs(\036)908 2221 y Fp(L)956 2202 y Fe(\003)996 2206 y Fw(\()p Fs(u)p Fw(\))27 b(=)1269 2133 y Fs(\032)1319 2052 y Fm(\000)1365 2133 y Fw(\()p Fs(L)1469 2096 y Fo(\003)1530 2133 y Ft(\\)c Fs(A)1692 2096 y Fo(\003)1732 2133 y Fw(\))f(+)g Fs(u)1946 2052 y Fm(\001)p 1269 2183 723 4 v 1514 2275 a Fs(\032)p Fw(\()p Fs(L)1668 2246 y Fo(\003)1708 2275 y Fw(\))2023 2206 y(+)2131 2133 y Fs(\032)2181 2052 y Fm(\000)2227 2133 y Fw(\()p Fs(L)2331 2096 y Fo(\003)2393 2133 y Ft(n)g Fs(A)2538 2096 y Fo(\003)2577 2133 y Fw(\))g(+)g Fs(u)2791 2052 y Fm(\001)p 2131 2183 706 4 v 2368 2275 a Fs(\032)p Fw(\()p Fs(L)2522 2246 y Fo(\003)2562 2275 y Fw(\))2847 2206 y Fs(:)-116 2420 y Fw(Since)33 b Fs(u)f Fw(is)g(p)s(erp)s(endicular)g(to)g(v)m(ectors)i(in)e Fs(L)1573 2384 y Fo(\003)1635 2420 y Ft(\\)22 b Fs(A)1796 2384 y Fo(\003)876 2608 y Fs(\032)926 2527 y Fm(\000)972 2608 y Fw(\()p Fs(L)1076 2572 y Fo(\003)1138 2608 y Ft(\\)g Fs(A)1299 2572 y Fo(\003)1339 2608 y Fw(\))g(+)g Fs(u)1553 2527 y Fm(\001)p 876 2659 723 4 v 1121 2750 a Fs(\032)p Fw(\()p Fs(L)1275 2721 y Fo(\003)1315 2750 y Fw(\))1636 2681 y(=)27 b Fs(e)1784 2640 y Fo(\000)p Fp(\031)r Fo(jj)p Fp(u)p Fo(jj)2003 2617 y Fh(2)2050 2614 y Fs(\032)p Fw(\()p Fs(L)2204 2578 y Fo(\003)2266 2614 y Ft(\\)c Fs(A)p Fw(\))p 2050 2659 416 4 v 2142 2750 a Fs(\032)p Fw(\()p Fs(L)2296 2721 y Fo(\003)2336 2750 y Fw(\))2504 2681 y Ft(\024)28 b Fw(0)p Fs(:)p Fw(135)p Fs(:)732 b Fw(\(2.10\))-116 2940 y(The)33 b(length)f(of)h(v)m(ectors)h(in)d(\()p Fs(L)1042 2904 y Fo(\003)1104 2940 y Ft(n)22 b Fs(A)1249 2904 y Fo(\003)1289 2940 y Fw(\))g(+)g Fs(u)32 b Fw(is)g(at)g(least) 1990 2901 y Fr(3)p 1990 2917 36 4 v 1990 2974 a(4)2036 2868 y Ft(p)p 2119 2868 59 4 v 72 x Fs(n)g Fw(and)h(hence)833 3142 y Fs(\032)883 3062 y Fm(\000)929 3142 y Fw(\()p Fs(L)1033 3106 y Fo(\003)1072 3142 y Fs(=)-5 b(A)1189 3106 y Fo(\003)1228 3142 y Fw(\))22 b(+)g Fs(u)1442 3062 y Fm(\001)p 833 3193 655 4 v 1044 3284 a Fs(\032)p Fw(\()p Fs(L)1198 3255 y Fo(\003)1238 3284 y Fw(\))1525 3216 y Ft(\024)1640 3142 y Fs(\032)1690 3062 y Fm(\000)1736 3142 y Fw(\()p Fs(L)1840 3106 y Fo(\003)1902 3142 y Fw(+)g Fs(u)p Fw(\))g Ft(n)2198 3103 y Fr(3)p 2198 3119 36 4 v 2198 3177 a(4)2243 3070 y Ft(p)p 2326 3070 59 4 v 72 x Fs(nB)2463 3062 y Fm(\001)p 1640 3193 869 4 v 1959 3284 a Fs(\032)p Fw(\()p Fs(L)2113 3255 y Fo(\003)2153 3284 y Fw(\))2547 3216 y Ft(\024)28 b Fw(0)p Fs(:)p Fw(285)p Fs(:)689 b Fw(\(2.11\))-360 3505 y(F)-8 b(rom)31 b (\(2.9\),\(2.10\),\(2.11\))g(w)m(e)i(obtain)f(0)p Fs(:)p Fw(43)27 b Ft(\024)h Fs(\036)1493 3520 y Fp(L)1541 3501 y Fe(\003)1581 3505 y Fw(\()p Fs(u)p Fw(\))f Ft(\024)h Fw(0)p Fs(:)p Fw(42,)k(a)h(con)m(tradiction.)1014 b Fi(\004)-360 3834 y Fn(2.6)161 b(Gaussian-lik)l(e)54 b(Measures)f(on)g(Lattices)-360 4053 y Fw(In)33 b(this)f(section)h(w)m(e)g(will)d(pro)m(v)m(e)k(Lemmas) e(2.5.3)g(and)g(2.5.2.)43 b(Our)33 b(only)f(to)s(ol)e(from)i(harmonic)f (analysis)h(on)g Fq(R)3787 4017 y Fp(n)-360 4174 y Fw(will)e(b)s(e)j (the)g(P)m(oisson)g(Summation)d(F)-8 b(orm)m(ula.)-360 4383 y Fu(Theorem)37 b(2.6.1)h(\(P)m(oisson)e(Summation)g(F)-9 b(orm)m(ula\))48 b Fj(L)-5 b(et)44 b Fs(f)54 b Fw(:)44 b Fq(R)2337 4347 y Fp(n)2434 4383 y Ft(!)g Fq(R)2644 4347 y Fp(n)2740 4383 y Fj(b)-5 b(e)44 b(a)f(c)-5 b(ontinuous)43 b(function,)-360 4503 y(such)35 b(that)g(for)g(some)f Fs(")27 b(>)h Fw(0)34 b Fj(and)g Fs(c)28 b Ft(2)g Fq(R)1230 4699 y Fs(f)11 b Fw(\()p Fs(x)p Fw(\))83 b Ft(\024)g Fs(c)p Fw(\(1)22 b(+)g Ft(j)p Fs(x)p Ft(j)p Fw(\))2061 4658 y Fo(\000)p Fp(n)p Fo(\000)p Fp(")3591 4699 y Fw(\(2.12\))1248 4829 y Fm(b)1230 4855 y Fs(f)10 b Fw(\()p Fs(x)p Fw(\))84 b Ft(\024)f Fs(c)p Fw(\(1)22 b(+)g Ft(j)p Fs(x)p Ft(j)p Fw(\))2061 4814 y Fo(\000)p Fp(n)p Fo(\000)p Fp(")3591 4855 y Fw(\(2.13\))-360 5061 y Fj(wher)-5 b(e)-67 5035 y Fm(b)-85 5061 y Fs(f)46 b Fj(is)34 b(the)h(F)-7 b(ourier)34 b(tr)-5 b(ansform)35 b(of)f Fs(f)46 b Fj(over)34 b Fq(R)1563 5025 y Fp(n)1616 5061 y Fj(.)45 b(Then)1301 5174 y Fm(X)1287 5386 y Fp(a)p Fo(2)p Fc(Z)1422 5367 y Fg(n)1475 5269 y Fs(f)11 b Fw(\()p Fs(a)p Fw(\))28 b(=)1806 5174 y Fm(X)1792 5386 y Fp(a)p Fo(2)p Fc(Z)1927 5367 y Fg(n)1998 5242 y Fm(b)1980 5269 y Fs(f)11 b Fw(\()p Fs(a)p Fw(\))p Fs(:)p eop %%Page: 33 37 33 36 bop -180 68 a Fk(2.6.)76 b(GA)m(USSIAN-LIKE)33 b(MEASURES)h(ON)f(LA)-8 b(TTICES)1807 b Fw(33)-180 400 y Fu(Pro)s(of)37 b(:)-180 520 y Fw(De\014ne)c(a)f(function)g Fs(F)42 b Fw(:)28 b Fq(T)808 484 y Fp(n)886 520 y Ft(!)f Fq(R)1462 728 y Fs(F)14 b Fw(\()p Fs(x)p Fw(\))28 b(=)1815 634 y Fm(X)1801 846 y Fp(a)p Fo(2)p Fc(Z)1936 827 y Fg(n)1989 728 y Fs(f)11 b Fw(\()p Fs(x)23 b Fw(+)f Fs(a)p Fw(\))p Fs(:)1393 b Fw(\(2.14\))-180 1031 y(The)38 b(righ)m(t-hand)d(side)i(of) g(\(2.14\))f(con)m(v)m(erges)j(uniformly)34 b(b)s(ecause)k(of)f (\(2.12\).)55 b(Clearly)36 b Fs(F)50 b Fw(is)36 b(con)m(tin)m(uous)i (and)-180 1151 y(in)m(tegrable.)k(W)-8 b(e)33 b(ha)m(v)m(e)-51 1377 y Fm(b)-69 1402 y Fs(F)13 b Fw(\()p Fs(y)t Fw(\))27 b(=)266 1267 y Fm(Z)321 1492 y Fc(T)371 1473 y Fg(n)428 1402 y Fs(F)14 b Fw(\()p Fs(x)p Fw(\))p Fs(e)681 1361 y Fo(\000)p Fr(2)p Fp(\031)r(iy)875 1338 y Fg(T)925 1361 y Fp(x)985 1402 y Fs(d)1036 1361 y Fp(n)1083 1402 y Fs(x)28 b Fw(=)1269 1267 y Fm(Z)1325 1492 y Fc(T)1375 1473 y Fg(n)1446 1308 y Fm(X)1432 1520 y Fp(a)p Fo(2)p Fc(Z)1567 1501 y Fg(n)1620 1402 y Fs(f)11 b Fw(\()p Fs(x)22 b Fw(+)g Fs(a)p Fw(\))p Fs(e)2026 1361 y Fo(\000)p Fr(2)p Fp(\031)r(iy)2220 1338 y Fg(T)2270 1361 y Fr(\()p Fp(x)p Fr(+)p Fp(a)p Fr(\))2478 1402 y Fs(d)2529 1361 y Fp(n)2575 1402 y Fs(x)28 b Fw(=)2762 1267 y Fm(Z)2817 1492 y Fc(R)2865 1473 y Fg(n)2928 1402 y Fs(f)11 b Fw(\()p Fs(x)p Fw(\))p Fs(e)3163 1361 y Fr(2)p Fp(\031)r(iy)3302 1338 y Fg(T)3352 1361 y Fp(x)3412 1402 y Fs(d)3463 1361 y Fp(n)3510 1402 y Fs(x)28 b Fw(=)3715 1376 y Fm(b)3697 1402 y Fs(f)10 b Fw(\()p Fs(y)t Fw(\))p Fs(:)-180 1727 y Fw(Condition)31 b(\(2.13\))h(giv)m(es)h(us)940 1702 y Fm(b)921 1727 y Fs(F)42 b Ft(2)28 b Fs(L)1186 1691 y Fr(1)1225 1727 y Fw(\()p Fq(Z)1332 1691 y Fp(n)1377 1727 y Fw(\).)43 b(Hence)34 b(for)e(an)m(y)h Fs(x)c Ft(2)f Fq(T)2349 1691 y Fp(n)2432 1727 y Fw(the)33 b(in)m(v)m(ersion)f(form)m(ula)f(holds)1397 1953 y Fs(F)14 b Fw(\()p Fs(x)p Fw(\))28 b(=)1750 1858 y Fm(X)1737 2070 y Fp(a)p Fo(2)p Fc(Z)1871 2051 y Fg(n)1942 1927 y Fm(b)1925 1953 y Fs(f)10 b Fw(\()p Fs(a)p Fw(\))p Fs(e)2155 1912 y Fo(\000)p Fr(2)p Fp(\031)r(ia)2349 1888 y Fg(T)2399 1912 y Fp(x)-180 2250 y Fw(and)33 b(for)f Fs(x)c Fw(=)f(0)33 b(w)m(e)g(obtain)f(the)h(result.)2635 b Fi(\004)-180 2465 y Fu(Lemma)37 b(2.6.2)49 b Fj(L)-5 b(et)35 b Fs(B)40 b Fj(b)-5 b(e)35 b(an)g Fs(n)22 b Ft(\002)h Fs(n)35 b Fj(matrix)f(and)g Fs(u)28 b Ft(2)g Fq(R)2056 2429 y Fp(n)2109 2465 y Fj(.)44 b(F)-7 b(or)34 b Fs(f)11 b Fw(\()p Fs(x)p Fw(\))28 b(=)g(exp)2852 2384 y Fm(\000)2920 2465 y Ft(\000)23 b Fw(\()p Fs(B)5 b(x)22 b Fw(+)g Fs(u)p Fw(\))3406 2429 y Fp(T)3461 2465 y Fw(\()p Fs(B)5 b(x)22 b Fw(+)g Fs(u)p Fw(\))3847 2384 y Fm(\001)706 2723 y(b)688 2749 y Fs(f)11 b Fw(\()p Fs(y)t Fw(\))27 b(=)1042 2682 y Fs(\031)1101 2646 y Fp(n=)p Fr(2)p 1015 2727 232 4 v 1015 2818 a Fw(det)17 b Fs(B)1273 2749 y Fw(exp)1438 2669 y Fm(\000)1506 2749 y Ft(\000)23 b Fs(\031)1665 2708 y Fr(2)1704 2749 y Fw(\()p Fs(B)1821 2708 y Fo(\000)p Fp(T)1931 2749 y Fs(y)t Fw(\))2021 2708 y Fp(T)2075 2749 y Fw(\()p Fs(B)2192 2708 y Fo(\000)p Fp(T)2302 2749 y Fs(y)t Fw(\))f(+)g(2)p Fs(\031)t(iu)2709 2708 y Fp(T)2763 2749 y Fw(\()p Fs(B)2880 2708 y Fo(\000)p Fp(T)2990 2749 y Fs(y)t Fw(\))3080 2669 y Fm(\001)3125 2749 y Fs(:)-180 2999 y Fu(Remark)37 b(3)49 b Fw(F)-8 b(or)31 b Fs(f)11 b Fw(\()p Fs(x)p Fw(\))28 b(=)g(exp)1011 2918 y Fm(\000)1079 2999 y Ft(\000)23 b Fs(\031)t Fw(\()p Fs(B)5 b(x)22 b Fw(+)g Fs(u)p Fw(\))1624 2963 y Fp(T)1679 2999 y Fw(\()p Fs(B)5 b(x)22 b Fw(+)h Fs(u)p Fw(\))2066 2918 y Fm(\001)2143 2999 y Fw(w)m(e)34 b(obtain)726 3235 y Fm(b)708 3262 y Fs(f)11 b Fw(\()p Fs(y)t Fw(\))26 b(=)1126 3194 y(1)p 1035 3239 V 1035 3330 a(det)17 b Fs(B)1293 3262 y Fw(exp)1458 3181 y Fm(\000)1526 3262 y Ft(\000)23 b Fs(\031)t Fw(\()p Fs(B)1802 3220 y Fo(\000)p Fp(T)1911 3262 y Fs(y)t Fw(\))2001 3220 y Fp(T)2056 3262 y Fw(\()p Fs(B)2173 3220 y Fo(\000)p Fp(T)2283 3262 y Fs(y)t Fw(\))e(+)h(2)p Fs(\031)t(iu)2689 3220 y Fp(T)2743 3262 y Fw(\()p Fs(B)2860 3220 y Fo(\000)p Fp(T)2970 3262 y Fs(y)t Fw(\))3060 3181 y Fm(\001)3105 3262 y Fs(:)-180 3495 y Fu(Pro)s(of)37 b(of)h(Lemma)f(2.6.2)-79 3846 y Fm(b)-97 3873 y Fs(f)11 b Fw(\()p Fs(y)t Fw(\))26 b(=)220 3737 y Fm(Z)275 3963 y Fc(R)323 3944 y Fg(n)387 3873 y Fw(exp)552 3792 y Fm(\000)620 3873 y Ft(\000)d Fw(\()p Fs(B)5 b(x)22 b Fw(+)g Fs(u)p Fw(\))1106 3832 y Fp(T)1161 3873 y Fw(\()p Fs(B)5 b(x)22 b Fw(+)g Fs(u)p Fw(\))g Ft(\000)h Fw(2)p Fs(\031)t(iy)1862 3832 y Fp(T)1916 3873 y Fs(x)1971 3792 y Fm(\001)2033 3873 y Fs(d)2084 3832 y Fp(n)2115 3873 y Fs(x)28 b Fw(=)-104 3996 y Fm(Z)-49 4221 y Fc(R)-1 4202 y Fg(n)62 4131 y Fw(exp)228 4051 y Fm(\000)296 4131 y Ft(\000)22 b Fw(\()p Fs(B)5 b(x)23 b Fw(+)f Fs(u)f Fw(+)i Fs(\031)t(iB)1035 4090 y Fo(\000)p Fp(T)1145 4131 y Fs(y)t Fw(\))1235 4090 y Fp(T)1289 4131 y Fw(\()p Fs(B)5 b(x)23 b Fw(+)f Fs(u)f Fw(+)h Fs(\031)t(iB)1928 4090 y Fo(\000)p Fp(T)2038 4131 y Fs(y)t Fw(\))g(+)g(2)p Fs(\031)t(iy)2441 4090 y Fp(T)2495 4131 y Fs(B)2574 4090 y Fo(\000)p Fr(1)2668 4131 y Fs(u)g Ft(\000)h Fs(\031)2905 4090 y Fr(2)2944 4131 y Fw(\()p Fs(B)3061 4090 y Fo(\000)p Fp(T)3171 4131 y Fs(y)t Fw(\))3261 4090 y Fp(T)3315 4131 y Fw(\()p Fs(B)3432 4090 y Fo(\000)p Fp(T)3542 4131 y Fs(y)t Fw(\))3632 4051 y Fm(\001)3693 4131 y Fs(d)3744 4090 y Fp(n)3774 4131 y Fs(x)29 b Fw(=)1827 4341 y Fs(\031)1886 4304 y Fp(n=)p Fr(2)p 1800 4385 V 1800 4476 a Fw(det)17 b Fs(B)2058 4408 y Fw(exp)2224 4327 y Fm(\000)2291 4408 y Ft(\000)23 b Fs(\031)2450 4367 y Fr(2)2489 4408 y Fw(\()p Fs(B)2606 4367 y Fo(\000)p Fp(T)2716 4408 y Fs(y)t Fw(\))2806 4367 y Fp(T)2860 4408 y Fw(\()p Fs(B)2977 4367 y Fo(\000)p Fp(T)3087 4408 y Fs(y)t Fw(\))f(+)g(2)p Fs(\031)t(iu)3494 4367 y Fp(T)3548 4408 y Fw(\()p Fs(B)3665 4367 y Fo(\000)p Fp(T)3775 4408 y Fs(y)t Fw(\))3865 4327 y Fm(\001)3910 4408 y Fs(:)3943 4640 y Fi(\004)-180 4878 y Fw(F)-8 b(or)32 b(notational)e(con)m(v)m(enience)35 b(w)m(e)e(de\014ne)h(for)e Fs(A)c Ft(\022)g Fq(R)1841 4842 y Fp(n)1521 5087 y Fs(\032)1571 5046 y Fo(0)1594 5087 y Fw(\()p Fs(A)p Fw(\))g(=)1875 4992 y Fm(X)1877 5204 y Fp(x)p Fo(2)p Fp(A)2035 5087 y Fs(e)2080 5046 y Fo(\000jj)p Fp(x)p Fo(jj)2255 5022 y Fh(2)2292 5087 y Fs(:)-180 5400 y Fw(Note)33 b(that)f Fs(\032)317 5364 y Fo(0)341 5400 y Fw(\()p Fs(A)p Fw(\))27 b(=)h Fs(\032)p Fw(\()p Fs(\031)768 5364 y Fo(\000)p Fr(1)p Fp(=)p Fr(2)933 5400 y Fs(A)p Fw(\))33 b(where)g Fs(\032)g Fw(is)f(de\014ned)i(b)m(y)g(\(2.8\).)p eop %%Page: 34 38 34 37 bop -360 68 a Fw(34)57 b Fk(CHAPTER)34 b(2.)76 b(HARMONIC)33 b(ANAL)-8 b(YSIS)34 b(O)m(VER)f(LOCALL)-8 b(Y)33 b(COMP)-8 b(A)m(CT)34 b(ABELIAN)g(GR)m(OUPS)-360 400 y Fu(Lemma)j(2.6.3)49 b Fj(F)-7 b(or)34 b(a)h(lattic)-5 b(e)35 b Fs(L)28 b Ft(\022)g Fq(R)1137 363 y Fp(n)1190 400 y Fj(,)35 b(any)f Fw(0)28 b Fs(<)f(t)h Ft(\024)h Fw(1)34 b Fj(and)g(any)h Fs(u)27 b Ft(2)i Fq(R)2493 363 y Fp(n)1288 638 y Fs(\032)1338 597 y Fo(0)1361 638 y Fw(\()p Fs(tL)23 b Fw(+)f Fs(u)p Fw(\))27 b Ft(\024)1874 571 y Fw(1)p 1857 615 83 4 v 1857 706 a Fs(t)1892 677 y Fp(n)1949 638 y Fs(\032)1999 597 y Fo(0)2023 638 y Fw(\()p Fs(L)p Fw(\))p Fs(:)-360 863 y Fu(Pro)s(of)37 b(:)-360 983 y Fw(Let)c Fs(B)k Fw(b)s(e)c(the)g(matrix)e(of)h(the)h (lattice)e Fs(L)p Fw(.)44 b(Using)32 b(the)h(P)m(oisson)g(summation)e (form)m(ula)-360 1243 y Fs(\032)-310 1202 y Fo(0)-286 1243 y Fw(\()p Fs(tL)p Fw(+)p Fs(u)p Fw(\))c(=)169 1149 y Fm(X)154 1360 y Fp(x)p Fo(2)p Fc(Z)291 1341 y Fg(n)344 1243 y Fw(exp)509 1163 y Fm(\000)555 1243 y Ft(\000)p Fw(\()p Fs(tB)5 b(x)p Fw(+)p Fs(u)p Fw(\))1009 1202 y Fp(T)1065 1243 y Fw(\()p Fs(tB)g(x)p Fw(+)p Fs(u)p Fw(\))1442 1163 y Fm(\001)1515 1243 y Fw(=)1705 1176 y Fs(\031)1764 1140 y Fp(n=)p Fr(2)p 1629 1220 331 4 v 1629 1312 a Fs(t)1664 1283 y Fp(n)1727 1312 y Fw(det)18 b Fs(B)1999 1149 y Fm(X)1985 1360 y Fp(y)r Fo(2)p Fc(Z)2120 1341 y Fg(n)2173 1243 y Fw(exp)2339 1133 y Fm(\020)2398 1243 y Ft(\000)2485 1176 y Fs(\031)2544 1140 y Fr(2)p 2485 1220 99 4 v 2497 1312 a Fs(t)2532 1283 y Fr(2)2594 1243 y Fw(\()p Fs(B)2711 1202 y Fo(\000)p Fp(T)2821 1243 y Fs(y)t Fw(\))2911 1202 y Fp(T)2965 1243 y Fw(\()p Fs(B)3082 1202 y Fo(\000)p Fp(T)3192 1243 y Fs(y)t Fw(\)+)3368 1176 y(2)p Fs(\031)t(i)p 3368 1220 141 4 v 3420 1312 a(t)3518 1243 y(y)3570 1202 y Fp(t)3599 1243 y Fs(B)3678 1202 y Fo(\000)p Fr(1)3772 1243 y Fs(u)3828 1133 y Fm(\021)3887 1243 y Fs(:)-360 1540 y Fw(Again)31 b(b)m(y)j(the)f(P)m(oisson)g(summation)d(form)m(ula) 46 1732 y(1)p 30 1777 83 4 v 30 1868 a Fs(t)65 1839 y Fp(n)122 1800 y Fs(\032)172 1759 y Fo(0)195 1800 y Fw(\()p Fs(L)p Fw(\))e(=)495 1732 y(1)p 479 1777 V 479 1868 a Fs(t)514 1839 y Fp(n)602 1705 y Fm(X)587 1917 y Fp(x)p Fo(2)p Fc(Z)725 1898 y Fg(n)778 1800 y Fw(exp)q(\()p Ft(\000)p Fw(\()p Fs(B)5 b(x)p Fw(\))1252 1759 y Fp(T)1308 1800 y Fw(\()p Fs(B)g(x)p Fw(\)\))28 b(=)1774 1732 y Fs(\031)1833 1696 y Fp(n=)p Fr(2)p 1697 1777 331 4 v 1697 1868 a Fs(t)1732 1839 y Fp(n)1796 1868 y Fw(det)17 b Fs(B)2067 1705 y Fm(X)2054 1917 y Fp(y)r Fo(2)p Fc(Z)2188 1898 y Fg(n)2241 1800 y Fw(exp)2407 1689 y Fm(\020)2489 1800 y Ft(\000)22 b Fs(\031)2647 1759 y Fr(2)2687 1800 y Fw(\()p Fs(B)2804 1759 y Fo(\000)p Fp(T)2914 1800 y Fs(y)t Fw(\))3004 1759 y Fp(T)3058 1800 y Fw(\()p Fs(B)3175 1759 y Fo(\000)p Fp(T)3285 1800 y Fs(y)t Fw(\))3375 1689 y Fm(\021)3433 1800 y Fs(:)-360 2096 y Fw(No)m(w)33 b(it)f(is)g(enough) h(to)f(notice)g(that)h(for)f(an)m(y)h Fs(y)e Ft(2)d Fq(Z)1587 2060 y Fp(n)130 2205 y Fm(\014)130 2265 y(\014)130 2324 y(\014)130 2384 y(\014)163 2349 y Fw(exp)328 2239 y Fm(\020)410 2349 y Ft(\000)520 2282 y Fs(\031)579 2246 y Fr(2)p 520 2326 99 4 v 531 2418 a Fs(t)566 2389 y Fr(2)628 2349 y Fw(\()p Fs(B)745 2308 y Fo(\000)p Fp(T)855 2349 y Fs(y)t Fw(\))945 2308 y Fp(T)999 2349 y Fw(\()p Fs(B)1116 2308 y Fo(\000)p Fp(T)1226 2349 y Fs(y)t Fw(\))21 b(+)1445 2282 y(2)p Fs(\031)t(i)p 1445 2326 141 4 v 1498 2418 a(t)1596 2349 y(y)1648 2308 y Fp(t)1677 2349 y Fs(B)1756 2308 y Fo(\000)p Fr(1)1850 2349 y Fs(u)1906 2239 y Fm(\021)1965 2205 y(\014)1965 2265 y(\014)1965 2324 y(\014)1965 2384 y(\014)2026 2349 y Ft(\024)28 b Fw(exp)2297 2239 y Fm(\020)2379 2349 y Ft(\000)22 b Fs(\031)2537 2308 y Fr(2)2577 2349 y Fw(\()p Fs(B)2694 2308 y Fo(\000)p Fp(T)2804 2349 y Fs(y)t Fw(\))2894 2308 y Fp(T)2948 2349 y Fw(\()p Fs(B)3065 2308 y Fo(\000)p Fp(T)3175 2349 y Fs(y)t Fw(\))3265 2239 y Fm(\021)3323 2349 y Fs(:)3763 2599 y Fi(\004)-360 2810 y Fu(Lemma)37 b(2.6.4)49 b Fj(F)-7 b(or)34 b(any)h(lattic)-5 b(e)35 b Fs(L)g Fj(any)g Fs(u)27 b Ft(2)h Fq(R)1505 2774 y Fp(n)1593 2810 y Fj(and)34 b Fs(c)28 b Ft(\025)1957 2725 y Fm(p)p 2057 2725 156 4 v 85 x Fs(n=)p Fw(2)804 3024 y Fs(\032)854 2987 y Fo(0)878 2943 y Fm(\000)924 3024 y Fw(\()p Fs(L)22 b Fw(+)g Fs(u)p Fw(\))g Ft(n)g Fs(cB)1457 2943 y Fm(\001)p 804 3074 699 4 v 1045 3166 a Fs(\032)1095 3137 y Fo(0)1119 3166 y Fw(\()p Fs(L)p Fw(\))1540 3097 y Ft(\024)1645 2957 y Fm(\022)1728 3030 y Fw(2)p Fs(c)1819 2994 y Fr(2)p 1728 3074 131 4 v 1764 3166 a Fs(n)1869 2957 y Fm(\023)1942 2977 y Fp(n=)p Fr(2)2076 3097 y Fw(exp)2242 2987 y Fm(\020)2311 3030 y Fs(n)p 2311 3074 59 4 v 2316 3166 a Fw(2)2401 3097 y Ft(\000)h Fs(c)2543 3056 y Fr(2)2583 2987 y Fm(\021)2659 3097 y Fs(:)-360 3349 y Fu(Pro)s(of)37 b(:)-360 3469 y Fw(Let)f Fs(x)e Ft(2)h Fw(\()p Fs(L)24 b Fw(+)h Fs(u)p Fw(\))f Ft(n)g Fs(cB)41 b Fw(and)c(let)e(0)e Fs(<)h(t)g Ft(\024)g Fw(1.)54 b(The)37 b(corresp)s(onding)f(elemen)m(t)g Fs(tx)e Ft(2)g Fs(t)p Fw(\()p Fs(L)26 b Fw(+)e Fs(u)p Fw(\))35 b(con)m(tributes)i(to)-360 3589 y Fs(\032)-310 3553 y Fo(0)-286 3509 y Fm(\000)-241 3589 y Fs(t)p Fw(\()p Fs(L)23 b Fw(+)f Fs(u)p Fw(\))113 3509 y Fm(\001)191 3589 y Fw(b)m(y)33 b(exp)q(\()p Ft(\000)p Fs(t)625 3553 y Fr(2)665 3589 y Ft(jj)p Fs(x)p Ft(jj)832 3553 y Fr(2)871 3589 y Fw(\).)43 b(W)-8 b(e)33 b(ha)m(v)m(e)582 3790 y(exp)q(\()p Ft(\000)p Fs(t)881 3754 y Fr(2)921 3790 y Ft(jj)p Fs(x)p Ft(jj)1088 3754 y Fr(2)1126 3790 y Fw(\))p 582 3835 583 4 v 619 3926 a(exp)q(\()p Ft(\000jj)p Fs(x)p Ft(jj)1050 3897 y Fr(2)1089 3926 y Fw(\))1202 3858 y(=)27 b(exp)1471 3777 y Fm(\000)1517 3858 y Fw(\(1)22 b Ft(\000)g Fs(t)1760 3817 y Fr(2)1800 3858 y Fw(\))p Ft(jj)p Fs(x)p Ft(jj)2005 3817 y Fr(2)2043 3777 y Fm(\001)2117 3858 y Ft(\025)28 b Fw(exp)2387 3777 y Fm(\000)2433 3858 y Fw(\(1)22 b Ft(\000)h Fs(t)2677 3817 y Fr(2)2716 3858 y Fw(\))p Fs(c)2796 3817 y Fr(2)2836 3777 y Fm(\001)2881 3858 y Fs(:)-360 4106 y Fw(Hence)714 4227 y Fs(\032)764 4186 y Fo(0)788 4146 y Fm(\000)833 4227 y Fw(\()p Fs(L)g Fw(+)f Fs(u)p Fw(\))f Ft(n)h Fs(cB)1366 4146 y Fm(\001)1440 4227 y Ft(\024)28 b Fs(\032)1595 4186 y Fo(0)1618 4146 y Fm(\000)1664 4227 y Fs(t)p Fw(\()p Fs(L)23 b Fw(+)f Fs(u)p Fw(\))2018 4146 y Fm(\001)2080 4227 y Fw(exp)2245 4146 y Fm(\000)2291 4227 y Fw(\()p Fs(t)2364 4186 y Fr(2)2426 4227 y Ft(\000)g Fw(1\))p Fs(c)2654 4186 y Fr(2)2693 4146 y Fm(\001)2739 4227 y Fs(:)-360 4392 y Fw(By)33 b(Lemma)e(2.6.3,)h Fs(\032)451 4356 y Fo(0)475 4312 y Fm(\000)521 4392 y Fs(t)p Fw(\()p Fs(L)22 b Fw(+)g Fs(u)p Fw(\))874 4312 y Fm(\001)947 4392 y Ft(\024)28 b Fw(\(1)p Fs(=t)p Fw(\))1261 4356 y Fp(n)1308 4392 y Fs(\032)1358 4356 y Fo(0)1382 4392 y Fw(\()p Fs(L)p Fw(\))k(and)h(hence)h(w)m(e)g(ha)m(v)m(e)737 4592 y Fs(\032)787 4551 y Fo(0)811 4511 y Fm(\000)857 4592 y Fw(\()p Fs(L)22 b Fw(+)g Fs(u)p Fw(\))g Ft(n)g Fs(cB)1390 4511 y Fm(\001)1463 4592 y Ft(\024)28 b Fw(\(1)p Fs(=t)p Fw(\))1777 4551 y Fp(n)1840 4592 y Fw(exp)2006 4511 y Fm(\000)2052 4592 y Fw(\()p Fs(t)2125 4551 y Fr(2)2186 4592 y Ft(\000)23 b Fw(1\))p Fs(c)2415 4551 y Fr(2)2454 4511 y Fm(\001)2500 4592 y Fs(\032)2550 4551 y Fo(0)2573 4592 y Fw(\()p Fs(L)p Fw(\))p Fs(:)-360 4791 y Fw(F)-8 b(or)32 b Fs(t)-150 4755 y Fr(2)-83 4791 y Fw(=)59 4752 y Fp(n)p 31 4768 101 4 v 31 4826 a Fr(2)p Fp(c)97 4807 y Fh(2)174 4791 y Fw(\(optimal)d(v)-5 b(alue\))32 b(w)m(e)i(obtain)d (the)i(result.)2023 b Fi(\004)-360 5018 y Fu(Corollary)36 b(2.6.5)i(\(of)f(Lemma)g(2.6.4\))49 b Fj(F)-7 b(or)34 b Fs(c)27 b Ft(\025)1653 4933 y Fm(p)p 1753 4933 291 4 v 85 x Fs(n=)p Fw(\(2)p Fs(\031)t Fw(\))757 5232 y Fs(\032)807 5151 y Fm(\000)853 5232 y Fw(\()p Fs(L)22 b Fw(+)g Fs(u)p Fw(\))g Ft(n)g Fs(cB)1386 5151 y Fm(\001)p 757 5282 675 4 v 998 5374 a Fs(\032)p Fw(\()p Fs(L)p Fw(\))1469 5305 y Ft(\024)1575 5165 y Fm(\022)1658 5238 y Fw(2)p Fs(\031)t(c)1808 5202 y Fr(2)p 1658 5282 190 4 v 1723 5374 a Fs(n)1857 5165 y Fm(\023)1930 5185 y Fp(n=)p Fr(2)2064 5305 y Fw(exp)2230 5195 y Fm(\020)2299 5238 y Fs(n)p 2299 5282 59 4 v 2304 5374 a Fw(2)2390 5305 y Ft(\000)g Fs(\031)t(c)2590 5264 y Fr(2)2630 5195 y Fm(\021)2706 5305 y Fs(:)p eop %%Page: 35 39 35 38 bop -180 68 a Fk(2.6.)76 b(GA)m(USSIAN-LIKE)33 b(MEASURES)h(ON)f(LA)-8 b(TTICES)1807 b Fw(35)-180 400 y Fu(Pro)s(of)37 b(of)h(Lemma)f(2.5.2)-180 520 y Fw(Plugging)31 b(in)h Fs(c)27 b Fw(=)525 481 y Fr(3)p 525 497 36 4 v 525 554 a(4)570 448 y Ft(p)p 653 448 59 4 v 72 x Fs(n)33 b Fw(in)m(to)f(Corollary)e(2.6.5)i(w)m(e)i(obtain)839 757 y Fs(\032)889 676 y Fm(\000)935 757 y Fw(\()p Fs(L)22 b Fw(+)g Fs(u)p Fw(\))g Ft(n)1357 717 y Fr(3)p 1357 734 36 4 v 1357 791 a(4)1402 685 y Ft(p)p 1485 685 59 4 v 72 x Fs(nB)1622 676 y Fm(\001)p 839 807 830 4 v 1157 899 a Fs(\032)p Fw(\()p Fs(L)p Fw(\))1706 830 y Ft(\024)1811 690 y Fm(\022)1894 763 y Fw(3)p 1894 807 49 4 v 1894 899 a(2)1953 830 y Fs(\031)t(e)2057 789 y Fr(1)p Fo(\000)2157 762 y Fh(9)p 2157 774 31 3 v 2157 815 a(8)2198 789 y Fp(\031)2245 690 y Fm(\023)2318 712 y Fp(n=)p Fr(2)2463 830 y Ft(\024)28 b Fw(\(0)p Fs(:)p Fw(285\))2867 789 y Fp(n=)p Fr(2)2984 830 y Fs(:)-180 1102 y Fw(whic)m(h)33 b(for)f Fs(n)c Ft(\025)g Fw(2)k(pro)m(v)m(es)j(2.5.2.)2893 b Fi(\004)-180 1341 y Fu(Pro)s(of)37 b(of)h(Lemma)f(2.5.3)-180 1461 y Fw(Let)c Fs(B)k Fw(b)s(e)c(the)g(matrix)e(of)h(the)h(lattice)e Fs(L)p Fw(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)618 1711 y Fs(\036)676 1726 y Fp(L)728 1711 y Fw(\()p Fs(u)p Fw(\))27 b(=)1001 1644 y Fs(\032)p Fw(\()p Fs(L)c Fw(+)f Fs(u)p Fw(\))p 1001 1688 369 4 v 1089 1779 a Fs(\032)p Fw(\()p Fs(L)p Fw(\))1407 1711 y(=)1592 1644 y(1)p 1520 1688 193 4 v 1520 1779 a Fs(\032)p Fw(\()p Fs(L)p Fw(\))1754 1616 y Fm(X)1739 1828 y Fp(x)p Fo(2)p Fc(Z)1876 1809 y Fg(n)1930 1711 y Fw(exp)2095 1630 y Fm(\000)2163 1711 y Ft(\000)h Fs(\031)t Fw(\()p Fs(B)5 b(x)22 b Fw(+)g Fs(u)p Fw(\))2708 1670 y Fp(T)2763 1711 y Fw(\()p Fs(B)5 b(x)22 b Fw(+)g Fs(u)p Fw(\))3149 1630 y Fm(\001)3194 1711 y Fs(:)550 b Fw(\(2.15\))-180 2019 y(By)33 b(Lemma)e(2.6.2)h(and)h (the)g(P)m(oisson)g(summation)e(form)m(ula)-97 2303 y(\(2)p Fs(:)p Fw(15\))c(=)489 2236 y(1)p 293 2280 441 4 v 293 2372 a Fs(\032)p Fw(\()p Fs(L)p Fw(\))17 b(det)h Fs(B)774 2209 y Fm(X)760 2420 y Fp(y)r Fo(2)p Fc(Z)895 2401 y Fg(n)948 2303 y Fw(exp)1114 2222 y Fm(\000)1182 2303 y Ft(\000)k Fs(\031)t Fw(\()p Fs(B)1457 2262 y Fo(\000)p Fp(T)1567 2303 y Fs(y)t Fw(\))1657 2262 y Fp(T)1711 2303 y Fw(\()p Fs(B)1828 2262 y Fo(\000)p Fp(T)1938 2303 y Fs(y)t Fw(\))2028 2222 y Fm(\001)2090 2303 y Fw(exp)2255 2222 y Fm(\000)2301 2303 y Fw(2)p Fs(\031)t(iu)2498 2262 y Fp(T)2552 2303 y Fw(\()p Fs(B)2669 2262 y Fo(\000)p Fp(T)2779 2303 y Fs(y)t Fw(\))2869 2222 y Fm(\001)2942 2303 y Fw(=)1250 2551 y(1)p 1055 2596 V 1055 2687 a Fs(\032)p Fw(\()p Fs(L)p Fw(\))17 b(det)g Fs(B)1534 2524 y Fm(X)1522 2735 y Fp(y)r Fo(2)p Fp(L)1654 2716 y Fe(\003)1706 2619 y Fw(exp)1872 2538 y Fm(\000)1940 2619 y Ft(\000)23 b Fs(\031)t Ft(jj)p Fs(y)t Ft(jj)2263 2577 y Fr(2)2300 2538 y Fm(\001)2362 2619 y Fw(exp)2528 2538 y Fm(\000)2595 2619 y Ft(\000)g Fw(2)p Fs(\031)t(iu)2892 2577 y Fp(T)2947 2619 y Fs(y)2999 2538 y Fm(\001)3071 2619 y Fw(=)3289 2551 y Fs(\032)p Fw(\()p Fs(L)3443 2515 y Fo(\003)3483 2551 y Fw(\))p 3185 2596 V 3185 2687 a Fs(\032)p Fw(\()p Fs(L)p Fw(\))17 b(det)g Fs(B)3657 2619 y Fm(c)-122 b Fs(\033)3690 2634 y Fp(L)3738 2615 y Fe(\003)3778 2619 y Fw(\()p Fs(u)p Fw(\))p Fs(:)-180 2935 y Fw(T)-8 b(o)33 b(\014nish)f(the)h(pro)s(of)f(note)87 3205 y Fs(\032)p Fw(\()p Fs(L)p Fw(\))c(=)425 3110 y Fm(X)410 3322 y Fp(x)p Fo(2)p Fc(Z)547 3303 y Fg(n)600 3205 y Fw(exp)766 3124 y Fm(\000)834 3205 y Ft(\000)23 b Fs(\031)t Fw(\()p Fs(B)5 b(x)p Fw(\))1203 3164 y Fp(T)1258 3205 y Fw(\()p Fs(B)g(x)p Fw(\))1468 3124 y Fm(\001)1542 3205 y Fw(=)1776 3137 y(1)p 1655 3182 291 4 v 1655 3273 a(det)q(\()p Fs(B)g Fw(\))1986 3110 y Fm(X)1972 3322 y Fp(y)r Fo(2)p Fc(Z)2107 3303 y Fg(n)2160 3205 y Fw(exp)2326 3124 y Fm(\000)2393 3205 y Ft(\000)23 b Fs(\031)t Fw(\()p Fs(B)2669 3164 y Fo(\000)p Fp(T)2779 3205 y Fs(y)t Fw(\))2869 3164 y Fp(T)2923 3205 y Fw(\()p Fs(B)3040 3164 y Fo(\000)p Fp(T)3150 3205 y Fs(y)t Fw(\))3240 3124 y Fm(\001)3312 3205 y Fw(=)3455 3137 y Fs(\032)p Fw(\()p Fs(L)3609 3101 y Fo(\003)3649 3137 y Fw(\))p 3426 3182 V 3426 3273 a(det\()p Fs(B)5 b Fw(\))3726 3205 y Fs(:)3943 3520 y Fi(\004)p -180 3641 4200 4 v -80 3782 a Fw(Maxim)m(um)31 b(of)h Fs(x)549 3746 y Fp(n)596 3782 y Fs(e)641 3746 y Fo(\000)p Fp(x)736 3722 y Fh(2)808 3782 y Fw(is)g(attained)f(for)h Fs(x)d Fw(=)1626 3697 y Fm(p)p 1726 3697 156 4 v 85 x Fs(n=)p Fw(2.)p eop %%Page: 36 40 36 39 bop -360 68 a Fw(36)57 b Fk(CHAPTER)34 b(2.)76 b(HARMONIC)33 b(ANAL)-8 b(YSIS)34 b(O)m(VER)f(LOCALL)-8 b(Y)33 b(COMP)-8 b(A)m(CT)34 b(ABELIAN)g(GR)m(OUPS)p eop %%Page: 37 41 37 40 bop -180 1064 a Fy(Chapter)78 b(3)-180 1479 y(Generalizations)e (of)i(Harmonic)f(Analysis)-180 1994 y Fn(3.1)161 b(In)l(tro)t(duction) -180 2213 y Fw(In)27 b(this)g(section)g(w)m(e)g(will)e(consider)i(the)g (theory)h(from)d(section)i(1)g(with)f(the)i(\014eld)e(of)h(complex)f(n) m(um)m(b)s(ers)i(replaced)-180 2333 y(b)m(y)j(a)f(\014nite)g(\014eld.) 43 b(Most)30 b(of)g(the)h(theorems)g(from)e(1)h(remain)e(v)-5 b(alid)29 b(ev)m(en)j(in)d(this)h(setting.)43 b(W)-8 b(e)30 b(omit)f(the)h(pro)s(ofs)-180 2453 y(b)s(ecause)k(they)f(are)g (iden)m(tical)e(with)h(those)h(in)f(section)h(1.)-80 2574 y(Let)d Fs(G)g Fw(b)s(e)g(a)f(\014nite)h(ab)s(elian)e(group,)j (let)e Fs(n)f Fw(=)f Ft(j)p Fs(G)p Ft(j)p Fw(.)42 b(Let)30 b Fs(t)g Fw(b)s(e)h(the)f(exp)s(onen)m(t)h(of)f Fs(G)g Fw(i.e.)42 b(the)30 b(smallest)f(p)s(ositiv)m(e)-180 2694 y(n)m(um)m(b)s(er)k(suc)m(h)h(that)e Fs(t)22 b Ft(\001)g Fs(g)31 b Fw(=)d(0)k(for)g(ev)m(ery)i Fs(g)d Ft(2)d Fs(G)p Fw(.)44 b(Let)32 b Fq(F)45 b Fw(b)s(e)32 b(the)h(\014nite)g(\014eld)f (with)g Fs(q)k Fw(elemen)m(ts)d(where)h Fs(t)p Ft(j)p Fs(q)26 b Ft(\000)c Fw(1.)-180 2815 y(Note)31 b(that)g(w)m(e)h(ha)m(v)m (e)g(primitiv)m(e)d Fs(t)p Fw(-th)i(ro)s(ots)f(of)h(unit)m(y)g(and)g(1) p Fs(=n)g Fw(in)f Fq(F)6 b Fw(.)49 b(\(T)-8 b(o)31 b(see)h(that)f(\()p Fs(n;)17 b(q)t Fw(\))27 b(=)h(1)i(note)h(that)g(an)m(y)-180 2935 y(prime)g Fs(p)i Fw(that)f(divides)h Fs(n)f Fw(divides)h Fs(t)g Fw(and)f(hence)i(do)s(es)f(not)g(divide)f Fs(q)t Fw(.\))-80 3055 y(W)-8 b(e)33 b(consider)f(the)h(space)h Fq(F)951 3019 y Fp(G)1049 3055 y Fw(of)e(functions)g(with)-35 3238 y Ft(\017)49 b Fw(p)s(oin)m(t)m(wise)32 b(m)m(ultiplication)c(\()p Fs(f)11 b(g)t Fw(\)\()p Fs(a)p Fw(\))27 b(=)g Fs(f)11 b Fw(\()p Fs(a)p Fw(\))p Fs(g)t Fw(\()p Fs(a)p Fw(\))p Fs(;)49 b(a)28 b Ft(2)g Fs(G)p Fw(,)-35 3435 y Ft(\017)49 b Fw(con)m(v)m(olution)32 b(\()p Fs(f)h Ft(\003)22 b Fs(g)t Fw(\)\()p Fs(x)p Fw(\))27 b(=)1142 3395 y Fr(1)p 1138 3412 43 4 v 1138 3469 a Fp(n)1225 3360 y Fm(P)1208 3522 y Fp(a)p Fo(2)p Fp(G)1364 3435 y Fs(f)11 b Fw(\()p Fs(a)p Fw(\))p Fs(g)t Fw(\()p Fs(x)21 b Ft(\000)i Fs(a)p Fw(\))p Fs(;)49 b(x)29 b Ft(2)f Fs(G)p Fw(,)k(and)-35 3671 y Ft(\017)49 b Fw(inner)32 b(pro)s(duct)h Ft(h)p Fs(f)5 b(;)17 b(g)t Ft(i)26 b Fw(=)1049 3632 y Fr(1)p 1045 3648 V 1045 3705 a Fp(n)1131 3596 y Fm(P)1114 3758 y Fp(a)p Fo(2)p Fp(G)1270 3671 y Fs(f)11 b Fw(\()p Fs(a)p Fw(\))p 1456 3584 178 4 v Fs(g)t Fw(\()p Fs(a)p Fw(\).)43 b(where)p 1986 3616 56 4 v 34 w Fs(x)28 b Fw(=)2173 3560 y Fm(n)2280 3610 y Fs(x)2335 3574 y Fo(\000)p Fr(1)2513 3610 y Fw(for)k Fs(x)c Ft(6)p Fw(=)g(0)2280 3730 y Fs(x)178 b Fw(for)32 b Fs(x)c Fw(=)g(0)2939 3671 y Fs(:)-80 3912 y Fw(Characters)g(are)f(homomorphisms)f(from)g Fs(G)h Fw(to)g(the)h(m)m(ultiplicativ)m(e)c(group)j(of)g Fq(F)5 b Fw(.)48 b(The)28 b(set)g(of)f(all)f(c)m(haracters)-180 4032 y(is)37 b(an)h(orthonormal)d(basis)i(of)g Fq(F)1042 3996 y Fp(G)1107 4032 y Fw(.)59 b(They)39 b(also)e(form)f(a)h(group)h (\(called)e(the)i(dual)f(group)g(of)g Fs(G)h Fw(o)m(v)m(er)g Fq(F)6 b Fw(\).)65 b(Fix)-180 4153 y Fs(G)28 b Fw(=)f Fq(Z)97 4168 y Fp(n)140 4177 y Fh(1)197 4153 y Ft(\010)21 b(\001)c(\001)g(\001)i(\010)i Fq(Z)599 4168 y Fp(n)642 4180 y Fg(k)714 4153 y Fw(and)31 b Fs(!)963 4168 y Fp(i)1023 4153 y Fw(b)s(e)h(a)g(primitiv)m(e)d Fs(n)1710 4168 y Fp(i)1739 4153 y Fw(-th)i(ro)s(ot)g(of)h(unit)m(y)g(in)f Fq(F)44 b Fw(\(since)32 b Fs(n)3008 4168 y Fp(i)3036 4153 y Ft(j)p Fs(t)g Fw(w)m(e)h(ha)m(v)m(e)g Fs(!)3559 4168 y Fp(i)3615 4153 y Ft(2)28 b Fq(F)6 b Fw(\).)49 b(F)-8 b(or)-180 4273 y Fs(b)28 b Fw(=)g(\()p Fs(b)72 4288 y Fr(1)112 4273 y Fs(;)17 b(:)g(:)g(:)e(;)i(b)371 4288 y Fp(k)414 4273 y Fw(\))28 b Ft(2)g Fs(G)k Fw(let)1560 4487 y Fs(\037)1621 4502 y Fp(b)1655 4487 y Fw(\()p Fs(x)p Fw(\))c(=)1962 4362 y Fp(k)1918 4392 y Fm(Y)1924 4602 y Fp(i)p Fr(=1)2062 4487 y Fs(!)2127 4442 y Fp(b)2157 4452 y Fg(i)2183 4442 y Fp(x)2223 4452 y Fg(i)2123 4512 y Fp(i)2253 4487 y Fs(:)-180 4733 y Fw(The)35 b(functions)f Fs(\037)505 4748 y Fp(b)574 4733 y Fw(are)g(c)m(haracters,)i(they)f (are)g(distinct)e(and)i(b)m(y)g(a)f(dimension)e(argumen)m(t)i(they)i (are)e(all)e(c)m(har-)-180 4854 y(acters)h(of)f Fs(G)p Fw(.)44 b(Hence)1503 5024 y Fm(b)1484 5049 y Fs(G)27 b Fw(=)1692 4939 y Fm(n)1758 5049 y Fs(\037)1819 5064 y Fp(b)1886 5049 y Ft(j)32 b Fs(b)d Ft(2)f Fs(G;)2263 4939 y Fm(o)2329 5049 y Fs(:)1463 b Fw(\(3.1\))-180 5280 y(An)m(y)35 b(function)e Fs(f)41 b Ft(2)30 b Fq(F)653 5243 y Fp(G)751 5280 y Fw(can)k(b)s(e)g(expressed)j(as)d(a)f(linear)g (com)m(bination)e(of)j(c)m(haracters.)48 b(The)35 b(co)s(e\016cien)m(t) f(of)g Fs(\037)3986 5295 y Fp(b)-180 5400 y Fw(is)e(denoted)i(b)m(y)440 5374 y Fm(b)422 5400 y Fs(f)11 b Fw(\()p Fs(\037)580 5415 y Fp(b)614 5400 y Fw(\).)44 b(The)33 b(function)1323 5374 y Fm(b)1305 5400 y Fs(f)39 b Fw(:)1465 5375 y Fm(b)1446 5400 y Fs(G)28 b Ft(!)f Fq(F)45 b Fw(is)32 b(called)f(the)i(F)-8 b(ourier)31 b(transform)h(of)g Fs(f)11 b Fw(.)1871 5649 y(37)p eop %%Page: 38 42 38 41 bop -360 68 a Fw(38)1181 b Fk(CHAPTER)34 b(3.)76 b(GENERALIZA)-8 b(TIONS)33 b(OF)f(HARMONIC)i(ANAL)-8 b(YSIS)-260 400 y Fw(Mapping)37 b Fs(b)g Ft(!)f Fs(\037)429 415 y Fp(b)501 400 y Fw(is)i(an)f(isomorphism)f(of)h Fs(G)g Fw(and)1765 374 y Fm(b)1746 400 y Fs(G)h Fw(and)g(hence)h(w)m(e) g(view)2729 373 y Fm(b)2711 400 y Fs(f)48 b Fw(as)38 b(a)g(function)f(in)g Fq(F)3580 363 y Fp(G)3683 400 y Fw(and)-360 520 y(write)-93 494 y Fm(b)-111 520 y Fs(f)11 b Fw(\()p Fs(b)p Fw(\))33 b(instead)f(of)563 494 y Fm(b)546 520 y Fs(f)10 b Fw(\()p Fs(\037)703 535 y Fp(b)738 520 y Fw(\).)-260 657 y(The)33 b(space)h Fq(F)270 604 y Ff(b)256 621 y Fp(G)353 657 y Fw(is)e(endo)m(w)m(ed)j(with)-215 874 y Ft(\017)49 b Fw(inner)32 b(pro)s(duct)h Ft(h)506 847 y Fw(^)546 874 y Fs(f)5 b(;)17 b(g)t Ft(i)682 847 y Fw(^)749 874 y(=)869 799 y Fm(P)851 960 y Fp(a)p Fo(2)p Fp(G)1007 874 y Fs(f)11 b Fw(\()p Fs(a)p Fw(\))p 1193 787 178 4 v Fs(g)t Fw(\()p Fs(a)p Fw(\),)32 b(and)-215 1154 y Ft(\017)49 b Fw(con)m(v)m(olution)32 b(\()p Fs(f)521 1151 y Fw(^)520 1154 y Ft(\003)17 b Fs(g)t Fw(\)\()p Fs(x)p Fw(\))27 b(=)954 1079 y Fm(P)937 1241 y Fp(a)p Fo(2)p Fp(G)1093 1154 y Fs(f)11 b Fw(\()p Fs(a)p Fw(\))p Fs(g)t Fw(\()p Fs(x)22 b Ft(\000)h Fs(a)p Fw(\).)-360 1429 y(By)33 b(the)g(orthogonalit)m(y)e(of)h(c)m(haracters)1060 1670 y Fm(b)1042 1696 y Fs(f)10 b Fw(\()p Fs(b)p Fw(\))28 b(=)g Ft(h)p Fs(f)5 b(;)17 b(\037)1546 1711 y Fp(b)1580 1696 y Ft(i)28 b Fw(=)1765 1628 y(1)p 1760 1673 59 4 v 1760 1764 a Fs(n)1845 1601 y Fm(X)1847 1813 y Fp(a)p Fo(2)p Fp(G)2005 1696 y Fs(f)11 b Fw(\()p Fs(a)p Fw(\))p 2191 1609 221 4 v Fs(\037)2252 1711 y Fp(a)2294 1696 y Fw(\()p Fs(b)p Fw(\))p Fs(:)1201 b Fw(\(3.2\))-360 2015 y Fu(Theorem)37 b(3.1.1)49 b Fj(The)34 b(F)-7 b(ourier)35 b(tr)-5 b(ansform)34 b(satis\014es)-215 2244 y Ft(\017)49 b Fj(line)-5 b(arity)289 2221 y Fq([)269 2244 y Fs(f)33 b Fw(+)22 b Fs(g)31 b Fw(=)647 2218 y Fm(b)629 2244 y Fs(f)i Fw(+)22 b Fm(b)-55 b Fs(g)s(;)942 2218 y Fm(c)937 2244 y Fs(af)38 b Fw(=)28 b Fs(a)1247 2218 y Fm(b)1229 2244 y Fs(f)11 b(;)51 b(f)5 b(;)17 b(g)31 b Ft(2)d Fq(F)1690 2208 y Fp(G)1756 2244 y Fs(;)51 b(a)28 b Ft(2)g Fq(C)-215 2473 y Ft(\017)-111 2447 y Fm(c)-116 2473 y Fs(f)11 b(g)31 b Fw(=)142 2447 y Fm(b)124 2473 y Fs(f)200 2470 y Fw(^)199 2473 y Ft(\003)17 b Fm(b)-55 b Fs(g)s(;)402 2450 y Fq([)395 2473 y Fs(f)33 b Ft(\003)22 b Fs(g)30 b Fw(=)747 2447 y Fm(b)729 2473 y Fs(f)11 b Fm(b)-55 b Fs(g)-215 2690 y Ft(\017)49 b(h)p Fs(f)5 b(;)17 b(g)t Ft(i)26 b Fw(=)i Ft(h)247 2664 y Fw(^)8 b Fm(b)287 2690 y Fs(f)i(;)17 b Fm(b)-55 b Fs(g)s Ft(i)427 2664 y Fw(^)467 2690 y Fs(:)34 b Fj(\(the)h(Plancher)-5 b(el)34 b(formula\))-360 3039 y Fn(3.2)161 b(Sums)52 b(of)i(matrix)g(columns)f(\(mo)t(d)g(m\))-360 3264 y Fw(W)-8 b(e)43 b(are)g(going)f(to)g(pro)m(v)m(e)j(a)d(Theorem)h (of)g(Th)m(\023)-46 b(erien)43 b([Th)m(\023)-46 b(e94)q(].)75 b(It)43 b(w)m(as)g(used)h(to)f(sho)m(w)h(that)f(a)f(circuit)g(with)-360 3384 y(MOD)-120 3399 y Fp(m)-21 3384 y Fw(gates)33 b(where)i Fs(m)e Fw(is)f(comp)s(osite)g(needs)i(at)f(least)g(\012\()p Fs(n)p Fw(\))g(gates)g(to)g(compute)g(the)g(AND)g(of)f Fs(n)h Fw(inputs.)45 b(A)-360 3505 y(MOD)-120 3520 y Fp(m)-21 3505 y Fw(gate)32 b(is)g(a)h(gate)f(whic)m(h)h(outputs)g(0)g (i\013)e(the)i(sum)g(of)f(its)g(inputs)g(is)g(divisible)f(b)m(y)i Fs(m)p Fw(.)p -194 3665 3869 4 v -194 4041 4 377 v -134 3776 a Fu(Theorem)38 b(3.2.1)49 b Fj(L)-5 b(et)44 b Fs(m;)17 b(s;)g(t)44 b Fj(b)-5 b(e)43 b(p)-5 b(ositive)43 b(inte)-5 b(gers.)72 b(If)43 b Fs(t)i(>)f(c)29 b Ft(\001)f Fs(s)h Ft(\001)f Fs(m)2668 3740 y Fr(11)p Fp(=)p Fr(2)2831 3776 y Fw(ln)16 b Fs(m)44 b Fj(wher)-5 b(e)43 b Fs(c)h Fj(is)f(an)-134 3896 y(absolute)d(c)-5 b(onstant,)41 b(then)f(for)f(any)h Fs(s)26 b Ft(\002)g Fs(t)41 b Fj(inte)-5 b(ger)39 b(matrix)h(ther)-5 b(e)40 b(exists)f(a)h(set)g(of)g(c)-5 b(olumns)39 b(which)-134 4017 y(sum)c(to)g(the)g(zer)-5 b(o)35 b(c)-5 b(olumn)34 b(mo)-5 b(dulo)34 b Fs(m)p Fj(.)p 3671 4041 V -194 4044 3869 4 v -260 4249 a Fw(W)-8 b(e)44 b(shall)e(use)i(the)g(follo)m(wing) d(explicit)h(v)m(ersion)i(of)f(Diric)m(hlet's)f(Theorem)i(ab)s(out)f (primes)g(in)g(arithmetic)-360 4369 y(progressions.)-360 4605 y Fu(Theorem)37 b(3.2.2)h(\([Hea90]\))48 b Fj(F)-7 b(or)41 b(any)h Fs(a)h Fj(c)-5 b(oprime)41 b(to)h Fs(m)g Fj(ther)-5 b(e)42 b(is)g(a)g(prime)g Fs(p)f Ft(\021)g Fs(a)h Fw(\()p Fj(mo)-5 b(d)34 b Fs(m)p Fw(\))43 b Fj(such)f(that)-360 4726 y Fs(p)28 b Ft(\024)g Fs(cm)-51 4690 y Fr(11)p Fp(=)p Fr(2)129 4726 y Fj(wher)-5 b(e)34 b Fs(c)h Fj(is)g(an)f(absolute)h(c)-5 b(onstant.)-360 4923 y Fw(The)25 b(estimate)e(in)g(Theorem)h(3.2.2)f (can)h(b)s(e)g(impro)m(v)m(ed)f(to)h(2\()p Fs(m)17 b Fw(ln)e Fs(m)p Fw(\))2176 4887 y Fr(2)2240 4923 y Fw(under)24 b(the)g(assumption)f(of)h(the)g(Extended)-360 5044 y(Riemann)31 b(Hyp)s(othesis)j([BS94)o(].)44 b(Theorem)33 b(3.2.1)f(follo)m(ws)f (from)h(Theorem)g(3.2.2)g(and)h(the)g(follo)m(wing)d(Lemma)-360 5280 y Fu(Lemma)37 b(3.2.3)49 b Fj(L)-5 b(et)40 b Fs(p)e Fj(b)-5 b(e)39 b(a)g(prime)f(and)g Fs(A)h Fj(b)-5 b(e)39 b(an)g Fs(s)25 b Ft(\002)g Fs(t)40 b Fj(inte)-5 b(ger)38 b(matrix.)57 b(If)38 b Fs(t)e(>)f(s)p Fw(\()p Fs(p)25 b Ft(\000)g Fw(1\))17 b(ln)o(\()p Fs(p)25 b Ft(\000)h Fw(1\))38 b Fj(then)-360 5400 y(ther)-5 b(e)35 b(exists)f(a)h(set)g(of) g(c)-5 b(olumns)34 b(which)g(sum)g(to)h(the)g(zer)-5 b(o)35 b(c)-5 b(olumn)34 b(mo)-5 b(dulo)34 b Fs(p)22 b Ft(\000)h Fw(1)p Fj(.)p eop %%Page: 39 43 39 42 bop -180 75 a Fk(3.2.)76 b(SUMS)33 b(OF)f(MA)-8 b(TRIX)34 b(COLUMNS)f(\(MOD)f(M\))1995 b Fw(39)-180 400 y(W)-8 b(e)38 b(use)g(harmonic)d(analysis)i(on)g(the)h(space)g(of)e (functions)i Fs(G)d Ft(!)g Fq(F)2391 415 y Fp(p)2474 400 y Fw(where)j Fs(G)e Fw(=)f Fq(Z)3053 363 y Fp(t)3053 424 y(p)p Fo(\000)p Fr(1)3180 400 y Fw(.)57 b(De\014ne)38 b(the)g(w)m(eigh)m(t)-180 520 y Fs(w)s(t)p Fw(\()p Fs(f)11 b Fw(\))37 b(of)g Fs(f)48 b Fw(as)37 b(the)h(n)m(um)m(b)s(er)g(of)f (non-zero)h(F)-8 b(ourier)36 b(co)s(e\016cien)m(ts)j(of)e Fs(f)11 b Fw(.)57 b(When)39 b(w)m(e)f(write)f Fs(f)49 b Fw(and)37 b Fs(g)k Fw(as)d(linear)-180 640 y(com)m(bination)30 b(of)j(c)m(haracters)g(w)m(e)h(see)-35 838 y Ft(\017)49 b Fs(w)s(t)p Fw(\()p Fs(f)32 b Ft(\001)22 b Fs(g)t Fw(\))27 b Ft(\024)h Fs(w)s(t)p Fw(\()p Fs(f)11 b Fw(\))22 b Ft(\001)f Fs(w)s(t)p Fw(\()p Fs(g)t Fw(\))-35 1039 y Ft(\017)49 b Fs(w)s(t)p Fw(\()p Fs(f)32 b Fw(+)22 b Fs(g)t Fw(\))27 b Ft(\024)h Fs(w)s(t)p Fw(\()p Fs(f)11 b Fw(\))22 b(+)g Fs(w)s(t)p Fw(\()p Fs(g)t Fw(\))-180 1237 y(F)-8 b(or)32 b(example)397 1215 y Fm(b)377 1237 y Fu(1)433 1252 y Fr(0)472 1237 y Fw(\()p Fs(x)p Fw(\))c(=)749 1197 y Fr(1)p 745 1214 43 4 v 745 1271 a Fp(n)830 1237 y Fw(for)k(an)m(y)h Fs(x)g Fw(and)g(hence)1539 1450 y Fs(w)s(t)p Fw(\()p Fu(1)1741 1465 y Fr(0)1780 1450 y Fw(\))28 b(=)f(\()p Fs(p)22 b Ft(\000)h Fw(1\))2245 1409 y Fp(t)2274 1450 y Fs(:)1518 b Fw(\(3.3\))-180 1663 y(Let)33 b Fs(B)g Fw(=)27 b Ft(f)p Fw(0)p Fs(;)17 b Fw(1)p Ft(g)447 1627 y Fp(t)503 1663 y Ft(\022)28 b Fq(Z)677 1627 y Fp(t)677 1687 y(p)p Fo(\000)p Fr(1)805 1663 y Fw(.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)951 1911 y Fm(c)943 1933 y Fu(1)999 1948 y Fp(B)1060 1933 y Fw(\()p Fs(x)p Fw(\))28 b(=)1337 1865 y(1)p 1332 1910 59 4 v 1332 2001 a Fs(n)1417 1838 y Fm(X)1418 2050 y Fp(y)r Fo(2)p Fp(B)1577 1933 y Fs(!)1642 1892 y Fp(x)1682 1868 y Fg(T)1730 1892 y Fp(y)1799 1933 y Fw(=)1917 1865 y(1)p 1913 1910 V 1913 2001 a Fs(n)1981 1933 y Fw(\(1)22 b(+)g Fs(!)2253 1892 y Fp(x)2293 1901 y Fh(1)2331 1933 y Fw(\))17 b Fs(:)g(:)g(:)e Fw(\(1)22 b(+)g Fs(!)2788 1892 y Fp(x)2828 1900 y Fg(t)2859 1933 y Fw(\))922 b(\(3.4\))-180 2250 y(Since)33 b(\(3.4\))f(is)g(non-zero)g (for)g(\()p Fs(p)23 b Ft(\000)f Fw(2\))1246 2214 y Fp(t)1308 2250 y Fw(c)m(hoices)33 b(of)g Fs(x)f Fw(w)m(e)i(obtain)1528 2463 y Fs(w)s(t)p Fw(\()p Fu(1)1730 2478 y Fp(B)1791 2463 y Fw(\))27 b(=)h(\()p Fs(p)22 b Ft(\000)g Fw(2\))2255 2422 y Fp(t)2285 2463 y Fs(:)1507 b Fw(\(3.5\))-180 2676 y Fu(Pro)s(of)37 b(of)h(Lemma)f(3.2.3)-180 2797 y Fw(Let)29 b Fs(A)f Fw(b)s(e)h(an)g Fs(s)14 b Ft(\002)g Fs(t)29 b Fw(in)m(teger)f(matrix)f(suc)m(h)j(that)e(no)h(set)g(of)f(columns)g (sums)h(to)f(the)h(zero)g(column)e(mo)s(dulo)g Fs(p)14 b Ft(\000)g Fw(1.)-180 2917 y(F)-8 b(or)32 b(eac)m(h)h(ro)m(w)g(of)f (the)h(matrix)e(de\014ne)j(a)e(function)1244 3130 y Fs(f)1292 3145 y Fp(i)1320 3130 y Fw(\()p Fs(x)p Fw(\))c(=)f Fs(!)1647 3089 y Fp(x)1687 3098 y Fh(1)1721 3089 y Fp(a)1758 3099 y Fg(i;i)1826 3089 y Fr(+)p Fo(\001\001\001)o Fr(+)p Fp(x)2035 3097 y Fg(t)2062 3089 y Fp(a)2099 3099 y Fg(i;t)2173 3130 y Fs(;)114 b(i)28 b Ft(2)g Fw([)p Fs(s)p Fw(])p Fs(:)-180 3343 y Fw(Consider)33 b(the)g(function)1374 3515 y Fs(f)38 b Fw(=)1611 3390 y Fp(s)1564 3420 y Fm(Y)1571 3630 y Fp(i)p Fr(=1)1708 3434 y Fm(\000)1754 3515 y Fw(1)21 b Ft(\000)i Fw(\(1)f Ft(\000)h Fs(f)2181 3530 y Fp(i)2209 3515 y Fw(\))2247 3473 y Fp(p)p Fo(\000)p Fr(1)2377 3434 y Fm(\001)2439 3515 y Fs(:)-180 3774 y Fw(If)32 b Fs(x)c Fw(=)g(0)j(then)i(eac)m(h)g Fs(f)673 3789 y Fp(i)701 3774 y Fw(\()p Fs(x)p Fw(\))28 b(=)g(1)k(and)g(hence)h Fs(f)11 b Fw(\()p Fs(x)p Fw(\))28 b(=)f(1.)43 b(Since)33 b(no)e(set)i(of)f(columns)f(sums)i(to)e(the)i(zero)f(column)-180 3894 y(mo)s(dulo)42 b Fs(p)29 b Ft(\000)i Fw(1,)46 b(for)d(an)m(y)i Fs(x)i Ft(2)g Fs(B)35 b Ft(n)29 b(f)p Fw(0)p Ft(g)44 b Fw(there)g(is)f Fs(i)h Fw(suc)m(h)h(that)f Fs(f)2389 3909 y Fp(i)2417 3894 y Fw(\()p Fs(x)p Fw(\))j Ft(6)p Fw(=)g(1)c(and)h(hence)h Fs(f)11 b Fw(\()p Fs(x)p Fw(\))47 b(=)g(0.)76 b(Th)m(us)-180 4015 y Fs(f)11 b Ft(j)-93 4030 y Fp(B)-5 4015 y Ft(\021)28 b Fu(1)156 4030 y Fr(0)195 4015 y Ft(j)223 4030 y Fp(B)284 4015 y Fw(.)-80 4135 y(Eac)m(h)42 b Fs(f)219 4150 y Fp(i)289 4135 y Fw(is)g(a)f(c)m (haracter.)72 b(The)43 b(function)e(1)28 b Ft(\000)h Fw(\(1)f Ft(\000)h Fs(f)2031 4150 y Fp(i)2059 4135 y Fw(\))2097 4099 y Fp(p)p Fo(\000)p Fr(1)2269 4135 y Fw(is)41 b(a)h(linear)e(com)m(bination)g(of)h(p)s(o)m(w)m(ers)i(of)f(the)-180 4255 y(c)m(haracter)48 b Fs(f)308 4270 y Fp(i)384 4255 y Fw(and)f(hence)i(has)f(w)m(eigh)m(t)g(at)f(most)g Fs(p)32 b Ft(\000)h Fw(1.)88 b(Hence)48 b(b)m(y)h(the)f(subm)m(ultiplicativit)m (y)c(of)j(w)m(eigh)m(t)-180 4376 y Fs(w)s(t)p Fw(\()p Fs(f)11 b Fw(\))27 b Ft(\024)h Fw(\()p Fs(p)22 b Ft(\000)h Fw(1\))491 4340 y Fp(t)520 4376 y Fw(.)-80 4496 y(Since)32 b Fs(f)h Ft(\001)22 b Fu(1)361 4511 y Fp(B)449 4496 y Fw(=)28 b Fu(1)609 4511 y Fr(0)876 4617 y Fw(\()p Fs(p)22 b Ft(\000)h Fw(1\))1172 4575 y Fp(t)1229 4617 y Fw(=)28 b Fs(w)s(t)p Fw(\()p Fu(1)1535 4632 y Fr(0)1574 4617 y Fw(\))f(=)h Fs(w)s(t)p Fw(\()p Fs(f)k Ft(\001)22 b Fu(1)2075 4632 y Fp(B)2136 4617 y Fw(\))27 b Ft(\024)h Fw(\()p Fs(p)23 b Ft(\000)f Fw(1\))2602 4575 y Fp(s)2639 4617 y Fw(\()p Fs(p)g Ft(\000)g Fw(2\))2934 4575 y Fp(t)-180 4788 y Fw(and)33 b(hence)1183 4955 y Fs(e)1228 4914 y Fp(t=)p Fr(\()p Fp(p)p Fo(\000)p Fr(1\))1502 4955 y Ft(\024)1607 4815 y Fm(\022)1680 4955 y Fw(1)22 b(+)1945 4888 y(1)p 1859 4932 220 4 v 1859 5024 a Fs(p)g Ft(\000)h Fw(2)2089 4815 y Fm(\023)2162 4837 y Fp(t)2219 4955 y Ft(\024)28 b Fw(\()p Fs(p)22 b Ft(\000)h Fw(1\))2620 4914 y Fp(s)-180 5184 y Fw(whic)m(h)33 b(yields)f Fs(t)c Ft(\024)g Fs(s)p Fw(\()p Fs(p)22 b Ft(\000)h Fw(1\))17 b(ln)o(\()p Fs(p)22 b Ft(\000)g Fw(1\).)2640 b Fi(\004)p eop %%Page: 40 44 40 43 bop -360 68 a Fw(40)1181 b Fk(CHAPTER)34 b(3.)76 b(GENERALIZA)-8 b(TIONS)33 b(OF)f(HARMONIC)i(ANAL)-8 b(YSIS)p eop %%Page: 41 45 41 44 bop -180 1064 a Fy(Bibliograph)-6 b(y)-180 1516 y Fw([A)m(C88])188 b(N.)46 b(Alon)f(and)h(F.R.K.)f(Ch)m(ung.)83 b(Explicit)45 b(construction)g(of)h(linear)e(sized)i(toleran)m(t)f(net) m(w)m(orks.)300 1637 y Fj(Discr)-5 b(ete)34 b(Mathematics)p Fw(,)f(72:15{19,)e(1988.)-180 1837 y([A)m(GHP92])49 b(N.)31 b(Alon,)f(O.)g(Goldreic)m(h,)g(J.)g(H)-12 b(\027)-61 b(astad,)32 b(and)e(R.)h(P)m(eralta.)39 b(Simple)29 b(construction)h (of)g(almost)f Fs(k)s Fw(-wise)300 1957 y(indep)s(enden)m(t)j(random)d (v)-5 b(ariables.)39 b Fj(R)-5 b(andom)31 b(Structur)-5 b(es)34 b(and)e(A)n(lgorithms)p Fw(,)e(3\(3\):289{304,)f(1992.)-180 2157 y([Ajt83])187 b(M.)34 b(Ajtai.)45 b(\006)810 2121 y Fr(1)810 2181 y(1)850 2157 y Fw(-form)m(ulae)32 b(on)h(\014nite)h (structures.)48 b Fj(A)n(nnals)35 b(of)g(Pur)-5 b(e)36 b(and)f(Applie)-5 b(d)36 b(L)-5 b(o)g(gic)p Fw(,)33 b(24:1{48,)300 2277 y(1983.)-180 2477 y([AKS83])125 b(M.)31 b(Ajtai,)e(J.)h(Koml\023) -49 b(os,)29 b(and)h(E.)g(Szemer)m(\023)-46 b(edi.)39 b(Sorting)29 b(in)g Fs(c)17 b Fw(log)f Fs(n)30 b Fw(parallel)d(steps.) 40 b Fj(Combinatoric)-5 b(a)p Fw(,)300 2597 y(3\(1\):1{19,)32 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