  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.
