Estimates of the best approximations and approximations of Fourier sums of classes of convolutions of periodic functions of not high smoothness in integral metrics

TL;DR

This paper derives precise estimates for the best possible approximations and Fourier sum approximations of certain classes of periodic convolution functions with low smoothness, within integral metric spaces.

Contribution

It provides exact order estimates for approximations of convolution classes with specific kernel coefficient conditions in $L_s$ spaces.

Findings

Exact order estimates for best approximations.

Exact order estimates for Fourier sum approximations.

Results apply to functions with kernels satisfying specific coefficient decay conditions.

Abstract

In metric of spaces $L_{s}, \ 1< s\leq\infty$, we obtain exact order estimates of best approximations and approximations by Fourier sums of classes of convolutions the periodic functions that belong to unit ball of space $L_{1}$, with generating kernel $\Psi_{\beta}(t)=\sum\limits_{k=1}^{\infty}\psi(k)\cos(kt-\frac{\beta\pi}{2})$, $\beta\in\mathbb{R}$, whose coefficients $\psi(k)$ are such that product $\psi(n)n^{1-\frac{1}{s}}$, $1<s\leq\infty$, can't tend to nought faster than every power function and besides, if $1<s<\infty$, then $\sum\limits_{k=1}^{\infty}\psi^{s}(k)k^{s-2}<\infty$ and if $s=\infty$, then $\sum\limits_{k=1}^{\infty}\psi(k)<\infty$.