This article is cited in 4 papers

Brief communications

Direct and inverse theorems for the approximation of functions by algebraic polynomials and splines in the norms of the Sobolev space

R. Z. Dautov

N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal University, 18 Kremlyovskaya str., Kazan, 420008 Russia

Abstract: In the one-dimensional case, interpolation weighted Besov spaces are defined, for functions from which direct and inverse estimates of the approximation error by algebraic polynomials and splines in Sobolev norms are valid. In a number of cases exact constants are indicated in the estimates. These results, as well as the inverse inequalities proved in the article, can be used to justify $p$- and $h$-$p$-finite element methods for solving boundary value problems for one-dimensional differential equations of order $2m$.

Keywords: Weighted Sobolev space, Besov interpolation space, direct and inverse approximation theorem, Bernstein inequality, inverse inequality.

UDC: 519.651

Received: 15.03.2022
Revised: 15.03.2022
Accepted: 08.04.2022

DOI: 10.26907/0021-3446-2022-6-79-86

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2022, 66:6, 65–72