Asymptotic estimates for the best uniform approximations of classes of convolution of periodic functions of high smoothness

TL;DR

This paper derives asymptotic estimates for the best uniform approximations of classes of convolutions of smooth periodic functions, providing precise bounds and conditions under which these estimates become exact.

Contribution

It introduces two-sided estimates for approximation errors of convolution classes with specific Fourier coefficient decay, advancing understanding of approximation of high-smoothness functions.

Findings

Established two-sided bounds for approximation errors.

Identified conditions for asymptotic equality of estimates.

Extended results to classes with particular Fourier coefficient decay.

Abstract

We find two-sides estimates for the best uniform approximations of classes of convolutions of $2\pi$-periodic functions from unit ball of the space $L_p, 1 \le p <\infty,$ with fixed kernels, modules of Fourier coefficients of which satisfy the condition $\sum\limits_{k=n+1}^\infty\psi(k)<\psi(n).$ In the case of $\sum\limits_{k=n+1}^\infty\psi(k)=o(1)\psi(n)$ the obtained estimates become the asymptotic equalities.