R. Z. Dautov, “Theorems on direct and inverse approximation by algebraic polynomials and piecewise polynomials in the spaces $H^m(a, b)$ and $B^s_{2,q}(a, b)$”, Izv. Vyssh. Uchebn. Zaved. Mat., 2024, Number 1,Pages 14

Theorems on direct and inverse approximation by algebraic polynomials and piecewise polynomials in the spaces $H^m(a, b)$ and $B^s_{2,q}(a, b)$

R. Z. Dautov

Kazan Federal University, 18 Kremlyovskaya str., Kazan, 420008 Russia

Abstract: The best estimates for the approximation error of functions, defined on a finite interval, by algebraic polynomials and piecewise polynomial functions are obtained in the case when the errors are measured in the norms of Sobolev and Besov spaces. We indicate the weighted Besov spaces, whose functions satisfy Jackson-type and Bernstein-type inequalities and, as a consequence, direct and inverse approximation theorems. In a number of cases, exact constants are indicated in the estimates.

Keywords: best approximation by polynomials, orthogonal polynomial, sharp error estimate, Bernstein's inequality, Jackson's inequality, direct and inverse theorems.

UDC: 517.587: 517.518

Received: 24.12.2022
Revised: 24.12.2022
Accepted: 29.03.2023

DOI: 10.26907/0021-3446-2024-1-14-34