Uniform approximations by Fourier sums on classes of convolutions of periodic functions

TL;DR

This paper derives asymptotic estimates for the best uniform approximation bounds of certain periodic functions represented as convolutions with fixed kernels, expanding understanding of Fourier sum approximations in function spaces.

Contribution

It provides new asymptotic estimates for uniform approximation bounds of convoluted periodic functions using Fourier sums, specifically for classes defined by convolution with fixed kernels.

Findings

Established asymptotic estimates for approximation bounds

Analyzed classes of functions represented by convolutions with fixed kernels

Extended Fourier approximation theory to new function classes

Abstract

We establish asymptotic estimates for exact upper bounds of uniform approximations by Fourier sums on the classes of $2\pi$-periodic functions, which are represented by convolutions of functions $\varphi (\varphi\bot 1)$ from unit ball of the space $L_{1}$ with fixed kernels $\Psi_{\beta}$ of the form $\Psi_{\beta}(t)=\sum\limits_{k=1}^{\infty}\psi(k) \cos\left(kt-\frac{\beta\pi}{2}\right)$, $\sum\limits_{k=1}^{\infty}k\psi(k)<\infty$, $\psi(k)\geq 0$, $\beta\in\mathbb{R}$.