Pointwise strong and very strong approximation by Fourier series of integrable functions

TL;DR

This paper investigates the approximation of integrable functions by Fourier series using generalized means, extending classical results with new estimates based on Gabisonia points and norm approximation.

Contribution

It introduces new approximation estimates for Fourier series of integrable functions, generalizing Totik, Marcinkiewicz, and Zygmund results using Gabisonia points.

Findings

Derived new bounds for Fourier series approximations

Extended classical approximation results to broader function classes

Provided norm approximation results for integrable functions

Abstract

We will present an estimation of the $H_{k_{0},k_{r}}^{q}f$ $\ $and $% H_{u}^{\lambda \varphi}f$ means as a approximation versions \ of \ the Totik type generalization$(\text{see \cite{11, 12}}) $ \ of the results of \ J. Marcinkiewicz and A. Zygmund in \cite{JM, ZA}. As a measure of such approximations we will use the function constructed on the base of definition of the Gabisonia points \cite{1}. Some results on the norm approximation will also given.