Abstract: In this paper we give a common generalization of the Walsh, Vilenkin system, the character system of the group of $2$-adic ($m$-adic) integers, the product system of normalized coordinate functions of 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 introduce the notion of the modulus of continuity on Vilenkin spaces, the concept of the best approximation by Vilenkin-like polynomials. We prove a Jackson type theorem.
Keywords: Vilenkin-like systems, best approximation, modulus of continuity, Jackson type inequality.
Classification (MSC2000): 42C10
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