Approximative characteristics of classes of functions $S^{\Omega}_{p,\theta}B(\mathbb{R}^d)$ with a given majorant of mixed modulus of smoothness

TL;DR

This paper derives order estimates for approximating functions within certain smoothness classes in $L_q$ spaces using entire functions of exponential type, based on the level surfaces of a specified function.

Contribution

It provides new order estimates for approximation in $L_q$ spaces for classes defined by mixed modulus of smoothness with a given majorant.

Findings

Order estimates of approximation are established.

Results apply to functions with mixed smoothness in $ ext{S}^ ext{ extOmega}_{p, heta}B$ classes.

Approximation uses entire functions with Fourier support in level surface sets.

Abstract

We obtain order estimates of approximation of functions from the classes $S^{\Omega}_{p,\theta}B (\mathbb{R}^d)$ in the space $L_q(\mathbb{R}^d)$, $1<p<q<\infty$, by entire functions of exponential type with supports of their Fourier transforms in sets generated by the level surfaces of a function $\Omega$.