Almost everywhere summability of Fourier series with indicating the set of convergence

TL;DR

This paper investigates conditions under which Fourier series linear means converge at points where the derivative of the integral of the function exists, providing criteria based on Wiener algebra membership.

Contribution

It establishes a criterion for the convergence of Fourier series linear means using multipliers and Wiener algebra conditions, extending understanding of convergence almost everywhere.

Findings

A criterion for convergence of $(C,1)$-means is established.

Sufficient conditions for convergence at points where the derivative exists are derived.

Examples illustrating the criteria are provided.

Abstract

The following problem is studied in this paper: Which multipliers $\{\lambda_{k, n}\}$ ensure the convergence, as $n\to \infty$, of the linear means of the Fourier series of functions $f\in L_1[-\pi, \pi]$ $$ \sum_{k=-\infty}^\infty \lambda_{k, n}\hat{f}_k e^{ikx}, $$ where $\widehat{f}_k$ is the $k$-th Fourier coefficient, at a point at which the derivative of the function $\int_0^x f$ exists. A criterion for the convergence of the $(C, 1)$-means ($\lambda_{k, n}=(1-\frac {|k|}{n+1})_+$) is found, while in the general case $\lambda_{k, n}=\phi(\frac {k}{n+1})$ a sufficient condition is derived for the convergence at all such points (that is, almost everywhere). The answer is given in terms of the belonging of $\phi(x)$ and $x\phi'(x)$ to the Wiener algebra of absolutely convergent Fourier integrals. The obtained results are supplemented by some examples.