Approximation of functions of several variables by linear methods in the space $S^p$

TL;DR

This paper investigates the approximation properties of specific operators in the space of multivariable periodic functions, providing estimates and characterizations of function classes based on Fourier series summation methods.

Contribution

It introduces and analyzes new summation operators for multiple Fourier series in $S^p$, offering constructive descriptions of function classes with generalized derivatives.

Findings

Operators $A^ riangle_{ ho,r}$ and $P^ riangle_{ ho,s}$ effectively approximate functions in $S^p$

Approximation estimates are established for these operators

Characterization of functions with derivatives in $S^pH_ ho$ classes

Abstract

In the spaces $S^p$ of functions of several variables, $2\pi$-periodic in each variable, we study the approximative properties of operators $A^\vartriangle_{\varrho,r}$ and $P^\vartriangle_{\varrho,s}$, which generate two summation methods of multiple Fourier series on triangular regions. In particular, in the terms of approximation estimates of these operators, we give a constructive description of classes of functions, whose generalized derivatives belong to the classes $S^pH_\omega$.