Pointwise Convergence of Fourier Series on the Ring of Integers of Local Fields with an Application to Gabor Systems

TL;DR

This paper investigates the convergence properties of Fourier series on the ring of integers of local fields, constructs examples of divergence, and applies these results to analyze Gabor systems and their basis properties.

Contribution

It provides new convergence results for Fourier series on local fields, establishes weighted maximal operator estimates, and characterizes Gabor system basis properties using $A_2$ weights and the Zak transform.

Findings

Constructed an integrable function with divergent Fourier series almost everywhere.

Proved pointwise convergence of Fourier series for $L^p$ functions with $A_p$ weights.

Characterized Schauder basis property of Gabor systems in local fields.

Abstract

We construct a simple example of an integrable function on the ring of integers of the $p$-adic field $\Q_p$ having an almost everywhere divergent Fourier series. On the other hand, we prove the pointwise convergence of the Fourier series of functions in $L^p(\D,w)$, $1<p<\infty$, where $\D$ is the ring of integers of a local field $K$ and $w$ is a weight in the Muckenhoupt $A_p$ class. This result includes, as special cases, when $\D$ is the ring of integers of $\Q_p$ or the field $\mathbb{F}_q((X))$ of formal Laurent series over a finite field $\mathbb{F}_q$, and in particular, when $\D$ is the Walsh-Paley or dyadic group $2^\omega$. To achieve this, we establish a weighted estimate for the maximal operator corresponding to the Fourier partial sum operators for functions in $L^p(\D,w)$. As an application, we characterize the Schauder basis property of the Gabor systems in a local field $K$ of positive characteristic in terms of the $A_2$ weights on $\D\times\D$ and the Zak transform $Zg$ of the window function $g$ that generates the Gabor system. Some examples are given to illustrate this result. In particular, we construct an example of a Gabor system which is complete and minimal, but fails to be a Schauder basis for $L^2(K)$.