Bernstein's theorem for compact groups

In this paper we generalize the classical Bernstein theorem concerning the absolute convergence of the Fourier series of Lipschitz functions. More precisely, we consider a group G which is finite dimensional, compact, and separable and has an infinite, closed, totally disconnected, normal subgroup D, such that $\text{G}\text{D}$ is a Lie group. Using this structure, we define in a natural way the notion of Lipschitz condition, and then prove that a function which satisfies a Lipschitz condition of order greater than $\text{(}\text{dim}\text{G + 1)}\text{2}$ belongs to the Fourier algebra of G.