ABSTRACT
This chapter presents an account of the convergence of Fourier series in Orlicz spaces on compact abelian groups, and especially contains an analysis of conjugate functions when the underlying space is a circle (torus) group G. We start with G = [0, 27r] and present Ryan's theorem that the conjugate function mapping on L*(G) is bounded iff the space is reflexive. Then we extend the result for certain subspaces (especially M$ determined by trigonometric polynomials) in L*(G) for a general class of TV-functions, $, and then give some extensions of these results for more general groups G. Only brief accounts of the latter can be included, since the necessary auxiliary results and applications are too numerous, demanding separate book length treatments.
