Summability of Multi-Dimensional Trigonometric Fourier Series

TL;DR

This paper investigates the summability and convergence properties of multi-dimensional trigonometric Fourier series, establishing boundedness of maximal operators and almost everywhere convergence for various summation methods.

Contribution

It proves boundedness of the maximal summation operator from Hardy space to Lp and establishes convergence results for multiple summability methods in higher dimensions.

Findings

Maximal operator of summability means is bounded from Hardy space Hp to Lp for p > p0.

Almost everywhere convergence of summability means for p=1.

Results extend to θ-summability and Fourier transforms.

Abstract

We consider the summability of one- and multi-dimensional trigonometric Fourier series. The Fej{\'e}r and Riesz summability methods are investigated in detail. Different types of summation and convergence are considered. We will prove that the maximal operator of the summability means is bounded from the Hardy space $H_p$ to $L_p$, for all $p>p_0$, where $p_0$ depends on the summability method and the dimension. For $p=1$, we obtain a weak type inequality by interpolation, which ensures the almost everywhere convergence of the summability means. Similar results are formulated for the more general $\theta$-summability and for Fourier transforms.