Generalized Lipschitz Spaces on Vilenkin Groups

Abstract

It is shown that, for certain choices of the defining indices, the generalized Lipschitz spaces on Vilenkin groups are included in certain Figa-Talamanca spaces Ap, and that the Fourier series of functions in the latter spaces converge uniformly. This result includes an extension of the classical Dini test, and endpoint versions of Bernstein's theorem on absolute convergence of Fourier series.