Almost everywhere convergence of Fourier series on compact connected Lie groups

TL;DR

This paper proves that Fourier series of Holder continuous functions on compact connected Lie groups converge almost everywhere, and identifies conditions under which divergence can occur on specific subsets.

Contribution

It establishes almost everywhere convergence of Fourier series for Holder continuous functions on compact Lie groups and constructs divergence examples on SU(2).

Findings

Fourier series of Holder continuous functions converge almost everywhere on G.

Convergence depends on the modulus of continuity of the function.

Divergence can occur on countable subsets of SU(2) for certain Holder functions.

Abstract

We consider the open problem: Does every square-integrable function f on a compact, connected Lie group G have an almost everywhere convergent Fourier series? We prove a general theorem from which it follows that if the integral modulus of continuity of f is O(t^a) for some a > 0 then the Fourier series of f converges almost everywhere on G. In particular, the Fourier series of any Holder continuous function of degree a > 0 on G converges almost everywhere. On the other hand, we show that to each countable subset E of G = SU(2) and each 0 < a < 1 there corresponds an Holder continuous function of degree a on SU(2) whose Fourier series diverges on E.