DIFFERENTIATION OF FUNCTIONS ON VILENKIN GROUPS

Abstract

In this paper we will consider a concept of differentiation of complex functions defined on Vilenkin groups, looking from the aspect of summability of Vilenkin series. In that sense we give matrix interpretation of C. N. Onneweer’s concept of differentiation [5, Definition 3], and then our generalization of that concept. We also give main properties of this generalized derivative. Moreover, we show that every method of summation of Vilenkin series, in a certain way, defines some concept of differentiation of integrable functions on Vilenkin groups.