Approximation of classes of convolutions of periodic functions by Zygmund sums in integral metrics

TL;DR

This paper derives precise estimates for how well Zygmund sums approximate classes of periodic functions in Lq spaces, based on their convolution representations with specific kernels, and identifies conditions where Zygmund sums achieve near-best approximation.

Contribution

It provides exact order estimates for Zygmund sums' deviations in Lq metrics for classes of periodic functions represented as convolutions, extending approximation theory results.

Findings

Zygmund sums achieve near-best approximation order for certain function classes.

Exact deviation estimates are obtained in Lq metrics for convolution-based classes.

Results specify parameter conditions for optimal approximation in periodic function classes.

Abstract

We obtain estimates exact in order for deviations of Zygmund sums in metrics of spaces $L_{q}$, $1<q<\infty$, on classes of $2\pi$-periodic functions, that admit the representation in the form of convolution of functions that belong to unit ball of the space $L_{1}$ with fixed kernel $\Psi_{\beta}$. We show that at certain values of the parameters that define the class $L^{\psi}_{\beta,1}$ and method of approximation, Zygmund sums provide the order of best approximation of given classes by trigonometric polynomials in metric $L_{q}$