Approximation of classes of convolutions of periodic functions by linear methods constructed on basis of Fourier-Lagrange coefficients

TL;DR

This paper determines the best possible pointwise and uniform approximation bounds for classes of periodic functions formed by convolutions, using linear polynomial methods based on Fourier-Lagrange coefficients.

Contribution

It introduces a method to calculate the least upper bounds for approximations of convolution classes using Fourier-Lagrange based linear polynomial methods.

Findings

Established bounds for pointwise approximation

Established bounds for uniform approximation

Applicable to classes of convolution-based periodic functions

Abstract

We calculate the least upper bounds of pointwise and uniform approximations for classes of $2\pi$-periodic functions expressible as convolutions of an arbitrary square summable kernel with functions, which belong to the unit ball of the space $L_2$, by linear polynomial methods constructed on the basis of their Fourier-Lagrange coefficients.