We give a common generalization of the Walsh system, Vilenkin system, the character system of the group of 2-adic (m-adic) integers, the product system of normalized coordinate functions for continuous irreducible unitary representations of the coordinate groups of noncommutative Vilenkin groups, the UDMD product systems (defined by F. Schipp) and some other systems. We prove that for integrable functions σₙf → f (n → ∞) a.e., where σₙf is the nth (C,1) mean of f. (For the character system of the...

C. Watari [12] obtained a simple characterization of Lipschitz classes $Li{p}^{\left(p\right)}\alpha \left(W\right)(1\ge p\ge \infty ,\alpha >0)$ on the dyadic group using the $L^p$-modulus of continuity and the best approximation by Walsh polynomials. Onneweer and Weiyi [4] characterized homogeneous Besov spaces $B_{p,q}^{\alpha }$ on locally compact Vilenkin groups, but there are still some gaps to be filled up. Our purpose is to give the characterization of Besov spaces $B_{p,q}^{\alpha }$ by oscillations, atoms and others on the dyadic groups. As applications, we show a strong capacity inequality of the...