A criterion for the density of algebraic polynomials in the spaces L_p (R,dμ), 1≤p<∞ (original) (raw)
Let μ be a positive Borel measure on R such that the polynomials belong to L_p (R,dμ). A necessary and sufficient condition is given for the polynomials to be dense in L_p (R,dμ). It reads as follows: the measure μ should be representable as dμ=w^p dv, with a finite Borel measure v and an upper semicontinuous function w:R→[0,1] such that the polynomials belong to and are dense in the space C_0^w={f∈C(R): f(x)w(x)→0 as |x|→∞}. The proof uses the Mergelyan's criterion for the density of polynomials in L_p spaces.
On the density principle for rational functions
Numerical Algorithms, 2000
Let E be a subspace of C(X) and define R(E):={g/h: g,heE;h>0}. We prove that R(E) is dense in C(X) if for every X0?X there exists xeX0 such that E contains an approximation to a d-function at the point x on the set X0. We use this principle to study the density of Müntz rationals in two variables.
On the size of lemniscates of polynomials in one and several variables
1996
In the convergence theory of rational interpolation and Pade approximation, it is essential to estimate the size of the lemniscatic set E := { z : |z| ≤ r and |P (z)| ≤ n } , for a polynomial P of degree ≤ n. Usually, P is taken to be monic, and either Cartan’s Lemma or potential theory is used to estimate the size of E, in terms of Hausdorff contents, planar Lebesgue measure m2, or logarithmic capacity cap. Here we normalize ‖P‖L∞ ( |z|≤r ) = 1 and show that cap(E) ≤ 2r and m2(E) ≤ π(2r )2 are the sharp estimates for the size of E. Our main result, however, involves generalizations of this to polynomials in several variables, as measured by Lebesgue measure on Cn or product capacity and Favarov’s capacity. Several of our estimates are sharp with respect to order in r and . §
A remark on the coefficients of bounded polynomials
Approximation Theory and Its Applications, 1996
Let p(x)= )', aj~ be such that Ip(e*)1~<1 for ~R and ]p(1) I=a6[O,1]. An inequality of Dewan J--O co~ for t~ sum I~1+ la.l,O<~u<v~n is 5ha~l~,~d. Let p(z) ----~'~a,z~ be such that lab(e")I~<1 for ~R, Ip(1) I=a6['0,1]. In 1-13,