On the Realization of Exact Upper Bounds of the Best Approximations on the Classes H1,1 by Favard Sums-Reference-Cited by-同舟云学术

On the Realization of Exact Upper Bounds of the Best Approximations on the Classes H1,1 by Favard Sums

Published:2023-08-02 Issue:8 Volume:12 Page:763
ISSN:2075-1680
Container-title:Axioms
language:en
Short-container-title:Axioms

Author:

Bushev Dmytro¹^ORCID,Kal’chuk Inna¹^ORCID

Affiliation:

1. Faculty of Information Technologies and Mathematics, Lesya Ukrainka Volyn National University, 43025 Lutsk, Ukraine

Abstract

In this paper, we find the sets of all extremal functions for approximations of the Hölder classes of H1 2π-periodic functions of one variable by the Favard sums, which coincide with the set of all extremal functions realizing the exact upper bounds of the best approximations of this class by trigonometric polynomials. In addition, we obtain the sets of all of extremal functions for approximations of the class H1 by linear methods of summation of Fourier series. Furthermore, we receive the set of all extremal functions for the class H1 in the Korneichuk–Stechkin lemma and its analogue, the Stepanets lemma, for the Hölder class H1,1 functions of two variables being 2π-periodic in each variable.

Publisher

MDPI AG

Subject

Geometry and Topology,Logic,Mathematical Physics,Algebra and Number Theory,Analysis

Link

https://www.mdpi.com/2075-1680/12/8/763/pdf

Reference14 articles.

1. A sharp estimate of the deviations of Favard sums over the classes H1,1A,B. Studies in the theory of approximation of functions and their applications;Stepanets;Akad. Nauk Ukrain SSR Inst. Mat. Kiev.,1978

2. Korneichuk, N.P. (1976). Extremal Problems in Approximation Theory, Nauka. (In Russian).

3. Inequalities of the type of Bernstein inequalities and their application to the investigation of the differential properties of the solutions of differential equations of higher order;Bushev;Dokl. Akad. Nauk USSR,1984

4. On Haar’s theorem concerning Chebysheff approximation problems heving unique solutions;Mairhuber;Proc. Am. Math. Soc.,1971

5. Stepanets, A.I. (1981). Uniform Approximations by Trigonometric Polynomials, Naukova Dumka. (In Russian); English translation: VSP: Leiden, The Netherland, 2001.