Multiple Fourier series and lattice point problems

TL;DR

This paper explores the convergence properties of multiple Fourier series of radial functions, revealing phenomena like Gibbs and Pinsky effects, and links these to lattice point problems and Hardy's conjecture in number theory.

Contribution

It establishes a connection between Fourier series convergence issues and classical lattice point problems, including Hardy's conjecture, for low dimensions.

Findings

Demonstrates the Gibbs-Wilbraham phenomenon in multiple Fourier series.

Identifies the Pinsky and third phenomena related to convergence.

Shows equivalence between Fourier series convergence and lattice point problems in 2D and 3D.

Abstract

For the multiple Fourier series of the periodization of some radial functions on $\mathbb{R}^d$, we investigate the behavior of the spherical partial sum. We show the Gibbs-Wilbraham phenomenon, the Pinsky phenomenon and the third phenomenon for the multiple Fourier series, involving the convergence properties of them. The third phenomenon is closely related to the lattice point problems, which is a classical theme of the analytic number theory. We also prove that, for the case of two or three dimension, the convergence problem on the Fourier series is equivalent to the lattice point problems in a sense. In particular, the convergence problem at the origin in two dimension is equivalent to Hardy's conjecture on Gauss's circle problem.