Uniform approximations by Fourier sums on classes of generalized Poisson integrals

TL;DR

This paper derives precise asymptotic bounds for Fourier sum approximations of periodic functions represented as convolutions with generalized Poisson kernels, providing explicit remainder estimates based on problem parameters.

Contribution

It introduces exact asymptotic equalities and explicit remainder estimates for uniform approximations of generalized Poisson integral classes by Fourier sums.

Findings

Asymptotic equalities for approximation bounds

Explicit remainder estimates in approximation

Application to classes of convolutions with Poisson kernels

Abstract

We find asymptotic equalities for exact upper bounds of approximations by Fourier sums in uniform metric on classes of $2\pi$-periodic functions, representable in the form of convolutions of functions $\varphi$, which belong to unit balls of spaces $L_{p}$, with generalized Poisson kernels. For obtained asymptotic equalities we introduce the estimates of remainder, which are expressed in the explicit form via the parameters of the problem.