Order estimates of the best approximations and approximations of Fourier sums of classes of convolutions of periodic functions of not high smoothness in uniform metric

TL;DR

This paper derives precise order estimates for the best uniform approximations and Fourier sums of convolutions of periodic functions with kernels having specific decay properties, focusing on functions with low smoothness.

Contribution

It provides exact order estimates for approximations of classes of convolutions with kernels characterized by specific Fourier coefficient decay, extending approximation theory for less smooth functions.

Findings

Exact order estimates for uniform approximations obtained.

Results apply to functions in $L_p$ spaces with specific kernel conditions.

Advances understanding of approximation limits for low-smoothness functions.

Abstract

We obtain exact for order estimates of best uniform approximations and uniform approximations by Fourier sums of classes of convolutions the periodic functions belong to unit balls of spaces $L_{p}, \ {1\leq p<\infty}$, with generating kernel $\Psi_{\beta}$, whose absolute values of Fourier coefficients $\psi(k)$ are such that $\sum\limits_{k=1}^{\infty}\psi^{p'}(k)k^{p'-2}<\infty$, $\frac{1}{p}+\frac{1}{p'}=1$, and product $\psi(n)n^{\frac{1}{p}}$ can't tend to nought faster than power functions.