Approximation of Nikol'skii-Besov functional classes $S^r_{1,\theta}B(\mathbb{R}^d)$ by entire functions of a special form

TL;DR

This paper derives precise estimates for approximating functions from Nikol'skii-Besov classes using entire functions of exponential type, focusing on the approximation error in the supremum norm.

Contribution

It provides the exact order of approximation for Nikol'skii-Besov classes by entire functions with spectrum restrictions, advancing understanding of function approximation in Lebesgue spaces.

Findings

Established exact-order approximation estimates in $L_$ norm.

Extended approximation theory to Nikol'skii-Besov classes with spectrum restrictions.

Provided new bounds for approximation errors of functions in these classes.

Abstract

We establish the exact-order estimates for the approximation of functions from the Nikol'skii-Besov classes $S^{\boldsymbol{r}}_{1,\theta} B(\mathbb{R}^d)$, $d\geqslant 1$, by entire function exponential type with some restrictions for their spectrum. The error of the approximation is estimated in the metric of the Lebesgue space $L_{\infty}(\mathbb{R}^d)$.