Almost everywhere convergence of Fourier series on SU(2): the case of Holder continuous functions

TL;DR

This paper investigates the convergence behavior of Fourier series for Holder continuous functions on SU(2), demonstrating that such series converge almost everywhere but can diverge on specific countable sets.

Contribution

It establishes that Fourier series of Holder continuous functions on SU(2) converge almost everywhere, and constructs functions diverging on any given countable set.

Findings

Fourier series of Holder continuous functions on SU(2) converge almost everywhere.

Existence of Holder continuous functions with divergent Fourier series on specified countable sets.

The results contribute to understanding the almost everywhere convergence problem for Fourier series on SU(2).

Abstract

We consider an aspect of the open problem: Does every square-integrable function on SU(2) have an almost everywhere convergent Fourier series? Let 0 < alpha < 1. We show that to each countable set E in SU(2) there corresponds an alpha-Holder continuous function on SU(2) whose Fourier series diverges on E. We also show that the Fourier series of each alpha-Holder continuous function on SU(2) converges almost everywhere.