On almost everywhere convergence of strong arithmetic means of Fourier series

TL;DR

This paper proves a real-variable method for Zygmund's theorem, showing almost everywhere convergence of strong arithmetic means of Fourier series, extending results to higher dimensions and broader function classes.

Contribution

It introduces a real-variable approach to Zygmund's theorem and generalizes the convergence results to functions on higher-dimensional tori.

Findings

Almost everywhere convergence of strong arithmetic means established

Extension of Zygmund's theorem to higher dimensions

Applicable to a broader class of functions on $ ext{T}^d$

Abstract

This article establishes a real-variable argument for Zygmund's theorem on almost everywhere convergence of strong arithmetic means of partial sums of Fourier series on $\mathbb{T}$, up to passing to a subsequence. Our approach extends to, among other cases, functions that are defined on $\mathbb{T}^d$, which allows us to establish an analogue of Zygmund's theorem in higher dimensions.