Approximation by Fourier sums on the classes of generalized Poisson integrals

TL;DR

This paper surveys asymptotic results for the uniform approximation errors of Fourier sums on classes of generalized Poisson integrals, focusing on the Kolmogorov--Nikolsky problem for these function classes.

Contribution

It provides a comprehensive overview of asymptotic equalities for the upper bounds of deviations of Fourier sums on generalized Poisson integral classes.

Findings

Derived asymptotic formulas for approximation errors.

Characterized the behavior of deviations for various parameters.

Connected Fourier sum deviations with generalized Poisson kernels.

Abstract

We present a survey of results related to the solution of Kolmogorov--Nikolsky problem for Fourier sums on the classes of generalized Poisson integrals $C^{\alpha,r}_{\beta,p}$, which consists in finding of asymptotic equalities for exact upper boundaries o f uniform norms of deviations of partial Fourier sums on the classes of $2\pi$--periodic functions $C^{\alpha,r}_{\beta,p}$, which are defined as convolutions of the functions, which belong to the unit balls pf the spaces $L_{p}$, $1\leq p\leq \infty$, with generalized Poisson kernels $$ P_{\alpha,r,\beta}(t)=\sum\limits_{k=1}^{\infty}e^{-\alpha k^{r}}\cos \big(kt-\frac{\beta\pi}{2}\big), \ \alpha>0, r>0, \ \beta\in \mathbb{R}.$$