Pointwise bounded approximation by polynomials (original) (raw)
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Uniform approximation by polynomials with integral coefficients. II
Pacific Journal of Mathematics, 1968
Let A be a discrete subring of C of rank 2. Let X be a compact subset of C with transfinite diameter not less than unity or with transfinite diameter less than unity, void interior, and connected complement. In an earlier paper we characterized the complex valued functions on X which can be uniformly approximated by elements from the ring of polynomials A[z], In this paper the same problem is studied where X is a compact subset of C with transfinite diameter d(X) less than unity and with nonvoid interior. It is also studied for certain compact subsets of C n where n is any positive integer. These subsets will have the property that every continuous function holomorphic on the interior is uniformly approximable by complex polynomials. A large class of sets of this type is shown to exist.
Ukrainian Mathematical Journal, 2007
We establish necessary and sufficient conditions under which a real-valued function from Lp(T), 1 ≤ p < ∞, is badly approximable by the Hardy subspace H 0 p := {f ∈ Hp : f (0) = 0}. In a number of cases, we obtain the exact values of the best approximations in the mean of functions holomorphic in the unit disk by functions holomorphic outside this disk. We use the obtained results for finding the exact values of the best polynomial approximations and n-widths of some classes of holomorphic functions. We establish necessary and sufficient conditions under which the generalized Bernstein inequality for algebraic polynomials on the unit circle is true.
Polynomial Approximation of Piecewise Analytic Functions
Journal of the London Mathematical Society, 1989
For a function / t h a t is piecewise analytic on [-1,1], we construct a sequence of polynomial approximants that converges to/at an exponential rate at each point of analyticity of/ For the uniform norm on [ -1,1], these polynomials approximate/to within a constant times the least possible error while, locally, the approximants give a ' best possible' rate of convergence. Moreover, unlike the best uniform approximants, the polynomials that we construct overconverge to an analytic continuation of/ Also, we prove a conjecture of Grothmann and Saff concerning the rate of polynomial approximation in a region of the plane to a complex extension of the absolute value function. As the starting point for our proofs, we obtain 'best possible' polynomial approximations to the sign function.
1999
2. The Weierstrass approximation theorem 3. Estimates for the Bernstein polynomials 4. Weierstrass' original proof 5. The Stone-Weierstrass approximation theorem 6. Chebyshev's theorems 7. Approximation by polynomials and trigonometric polynomials 8. The nonexistence of a continuous linear projection 9. Approximation of functions of higher regularity 10. Inverse theorems References Introductory remarks These notes comprise the main part of a course on approximation theory presented at Uppsala University in the Fall of 1998, viz. the part on polynomial approximation. The material is mainly classical. As sources I used Cheney [1966], Dzjadyk [1977], Korovkin [1959], and Lorentz [1953], as well as papers listed in the bibliography. The emphasis is on explaining the main ideas behind the most important techniques. The last part of the course was on rational approximation and is not included here. I followed mainly Cheney [1966, Chapter 5, pp. 150-167]. I also discussed Padé approximation briefly, following Cheney [1966, Chapter 5, pp. 173-177] and the introduction in Rudälv [1998]. I am grateful to Tsehaye Kahsu Araaya for remarks to the manuscript. 2 a k 2 − d(0, A) 2 , since 1 2 a j + 1 2 a k belongs to A in view of the convexity. We see that the right-hand side tends to zero as j, k → ∞. This implies that (a j) is a Cauchy sequence, and it must therefore have a limit in H. The limit cannot depend on the sequence, for if we take two sequences (a j) and (b j) and mix them, the new sequence (a 0 , b 0 , a 1 , b 1 , a 2 , ...) must converge by the same argument. We call the limit π(0); by translation we define π(x) ∈ A. Exercise 1.3. Prove that the mapping π: H → A is continuous; more precisely that π(x) − π(y) x − y , x, y ∈ H. Prove also that the set A is contained in a half-space as soon as x / ∈ A: in the real case every a ∈ A must satisfy (a − x|π(x) − x) π(x) − x 2. What about the complex case? Exercise 1.4. Prove that if A is a closed linear subspace, then x−π(x) is orthogonal to π(x). Prove that we get two idempotent mappings π and I − π, and determine all possible relations between the subspaces ker π, ker(I − π), im π, im(I − π). So Hilbert space is an easy case where the best approximant is unique. However, there are other interesting cases when we can prove uniqueness of the best approximant. Now we may perhaps dare to say that approximation theory is the study of approximating sequences, best approximants and their uniqueness or nonuniqueness in cases where X is a space of interesting functions, and A is some subspace of nice functions, like polynomials, trigonomentric polynomials,... 2. The Weierstrass approximation theorem The best starting point for these lectures is the classical Weierstrass 1 approximation theorem. It says that for any continuous real-valued function f on the interval [0, 1] and any integer k 1 there is a polynomial p k such that |f (x) − p k (x)| 1/k for all
Approximation by polynomials with bounded coefficients
Journal of Number Theory, 2007
Let θ be a real number satisfying 1 < θ < 2, and let A(θ) be the set of polynomials with coefficients in {0, 1}, evaluated at θ. Using a result of Bugeaud, we prove by elementary methods that θ is a Pisot number when the set (A(θ) − A(θ) − A(θ)) is discrete; the problem whether Pisot numbers are the only numbers θ such that 0 is not a limit point of (A(θ) − A(θ)) is still unsolved. We also determine the three greatest limit points of the quantities inf{c, c > 0, c ∈ C(θ)}, where C(θ) is the set of polynomials with coefficients in {−1, 1}, evaluated at θ , and we find in particular infinitely many Perron numbers θ such that the sets C(θ) are discrete.
Best Polynomial Approximation in -Norm and -Growth of Entire Functions
Abstract and Applied Analysis, 2013
The classical growth has been characterized in terms of approximation errors for a continuous function on by Reddy (1970), and a compact of positive capacity by Nguyen (1982) and Winiarski (1970) with respect to the maximum norm. The aim of this paper is to give the general growth (-growth) of entire functions in by means of the best polynomial approximation in terms of -norm, with respect to the set , where is the Siciak's extremal function on an -regular nonpluripolar compact is not pluripolar.
Interpolatory Pointwise Estimates for Polynomial Approximation X1. Introduction and Main Results
2007
We discuss whether or not it is possible to have interpolatory pointwise estimates in the approximation of a function f 2 C 0; 1], by polynomials. For the sake of completeness as well as in order to strengthen some existing results, we discuss brieey the situation in unconstrained approximation. Then we deal with positive and monotone constraints where we show exactly when such interpolatory estimates are achievable by proving aarmative results and by providing the necessary counterexamples in all other cases. The eeect of the endpoints of the nite interval on the quality of approximation of continuous functions by algebraic polynomials, was rst observed by Nikolski Nik46]. Later pointwise estimates of this phenomenon were given by Timan Tim51] (k = 1), Dzjadyk Dzj58, Dzj77] (k = 2), Freud Fre59] (k = 2), and Brudny Bru63] (k 2), who proved that if f 2 C r 0; 1], then for each n N = r + k ? 1, a polynomial p n 2 n exists, such that (1.1) jf(x) ? p n (x)j c(r; k) r n (x)! k (f (r) ; ...
On the Zeros of Polynomials of Best Approximation
Journal of Approximation Theory, 1999
Given a function f, uniform limit of analytic polynomials on a compact, regular set E/C N , we relate analytic extension properties of f to the location of the zeros of the best polynomial approximants to f in either the uniform norm on E or in appropriate L q norms. These results give multivariable versions of one-variable results due to Blatt Saff, Ples niak and Wo jcik.