Approximation by Fourier sums in classes of Weyl--Nagy differentiable functions with high exponent of smoothness

TL;DR

This paper derives asymptotic bounds for Fourier sum approximations of highly smooth Weyl--Nagy differentiable functions in uniform and Lp metrics, revealing how smoothness influences approximation quality.

Contribution

It provides new asymptotic estimates for Fourier approximation errors in classes of highly smooth Weyl--Nagy functions, especially for large smoothness exponents.

Findings

Asymptotic bounds for uniform approximation errors established.

Similar estimates derived for Lp spaces with p=1 to infinity.

Results highlight the impact of high smoothness on approximation accuracy.

Abstract

We establish asymptotic estimates for the least upper bounds of approximations in the uniform metric by Fourier sums of order $n-1$ of classes of $2\pi$-periodic Weyl--Nagy differentiable functions, $W^r_{\beta,p}, 1\le p\le \infty, \beta\in\mathbb{R},$ for high exponents of smoothness $r\ (r-1\ge \sqrt{n})$. We obtain similar estimates in metrics of the spaces $L_p, 1\le p\le\infty,$ for functional classes $W^r_{\beta,1}$.