Best approximations for classes of periodic functions of many variables with bounded dominating mixed derivative

TL;DR

This paper derives precise approximation estimates for multivariable periodic Sobolev functions with bounded mixed derivatives, using hyperbolic cross trigonometric polynomials, with error bounds in a specific function space.

Contribution

It provides exact order estimates for approximating Sobolev classes of multivariable periodic functions with bounded mixed derivatives using hyperbolic cross spectra.

Findings

Exact order estimates for approximation errors.

Use of hyperbolic cross trigonometric polynomials.

Error bounds in the space $B_{q,1}( ext{T}^d)$.

Abstract

We established exact in order estimates an approximation of the Sobolev classes $W^{\boldsymbol{r}}_{p,\boldsymbol{\alpha}}(\mathbb{T}^d)$ of periodic functions of many variables with a bounded dominating mixed derivative. The approximation is made using trigonometric polynomials with the spectrum in step-hyperbolic crosses, and the error is estimated in the metric of the space $B_{q,1}(\mathbb{T}^d)$, $1 \leqslant p, q < \infty$.