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 $\text{G}$G. 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 Lip_α(G) ? $\text{R}$(G). On certain groups this condition becomes: $\text{α >}\text{1}\text{2}$ (Bernstein's theorem). We then give a similar condition on α which if satisfied implies that Lip_α(G) ∈ BV(G) ? $\text{R}$(G). On certain groups this condition becomes: α > 0 (Zygmund's theorem). Moreover we show that $\text{α >}\text{1}\text{2}$ is best possible by showing that $\text{Lip}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(G)}$ ? $\text{R}$(G).