Order estimates of the best orthogonal trigonometric approximations of classes of convolutions of periodic functions of not high smoothness

TL;DR

This paper derives order estimates for the best uniform orthogonal trigonometric approximations of periodic functions with specific derivative properties, extending results to various function spaces and conditions on the function a(k).

Contribution

It provides new order estimates for approximations of classes of convolutions of periodic functions with low smoothness, including cases with slow decay of a(k) and summability conditions.

Findings

Order estimates established for uniform orthogonal trigonometric approximations.

Analogous estimates derived in L_s spaces for summable a,b-differentiable functions.

Results extend approximation theory to functions with minimal smoothness constraints.

Abstract

We obtain order estimates for the best uniform orthogonal trigonometric approximations of $2\pi$-periodic functions, whose $(\psi,\beta)$-derivatives belong to unit balls of spaces $L_{p}, \ 1\leq p<\infty$, in case at consequences $\psi(k)$ are that product $\psi(n)n^{\frac{1}{p}}$ can tend to zero slower than any power function and $\sum\limits_{k=1}^{\infty}\psi^{p'}(k)k^{p'-2}<\infty$ when $1<p<\infty$, $\frac{1}{p}+\frac{1}{p'}=1$ and $\sum\limits_{k=1}^{\infty}\psi(k)<\infty$ when $p=1$. We also establish the analogical estimates in $L_{s}$-metric, $1< s\leq \infty$, for classes of the summable $(\psi,\beta)$-differentiable functions, such that $\parallel f_{\beta}^{\psi}\parallel_{1}\leq1$.