SMOOTHNESS AND ABSOLUTE CONVERGENCE OF FOURIER-SERIES IN COMPACT TOTALLY DISCONNECTED GROUPS

In this paper we study in the context of compact totally disconnected groups the relationship between the smoothness of a function and its membership in the Fourier algebra GG. Specifically, we define a notion of smoothness which is natural for totally disconnected groups. This in turn leads to the notions of Lipshitz condition and bounded variation. We then give a condition on α which if satisfied implies Lipα(G) ⊂ R(G). On certain groups this condition becomes: α > 1 2 (Bernstein's theorem). We then give a similar condition on α which if satisfied implies that Lipα(G) ∈ BV(G) ⊂ R(G). On certain groups this condition becomes: α > 0 (Zygmund's theorem). Moreover we show that α > 1 2 is best possible by showing that Lip 1 2(G) ⊄ R(G). © 1978.