# Pointwise strong (H, Phi) approximation by Fourier series of L^{Psi} integrable functions

## TL;DR

This paper extends classical results on the strong summability of Fourier series by providing new approximation estimates for functions in L^{Psi} spaces, generalizing Hardy and Littlewood's work with a focus on generalized strong means.

## Contribution

It introduces an improved estimation of generalized strong means for Fourier series of L^{Psi} functions, extending classical summability results to broader function spaces.

## Key findings

- New approximation bounds for Fourier series in L^{Psi} spaces
- Extension of Hardy and Littlewood's classical results
- Corollaries and remarks on approximation measures

## Abstract

We essentially extend and improve the classical result of G. H. Hardy and J. E. Littlewood on strong summability of Fourier series. We will present an estimation of the generalized strong mean (H; Phi) as an approximation version of the Totik type generalization of the result of G. H. Hardy, J. E. Littlewood, in case of integrable functions from L^{Psi}. As a measure of such approximation we will use the function constructed by function Psi complementary to Phi on the base of defnition of the L^{Psi} points. Some corollary and remarks will also be given.