Almost everywhere convergence of Fourier series on compact connected Lie groups

2021/08/30 by Grow, David, Myers, Donnie
#22E30 #43A50 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2108.12997

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.

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