Uniform convergence and A^\lambda-boundedness ofseries with respect to product systems

Abstract

Let {g_k} be an orthogonal product system. For a continuous function u it is proved that the series ∑_k<u,w_k>g_k(t) with  the Walsh-Fourier coefficients <u,w_k> is convergent (A-summable, A^λ-bounded, regularly A^λ-summable) uniformly if and only if the Walsh-Fourier series ∑_k<u,w_k>w_k(t) has the same property.