Fourier multiplier theorem for atomic Hardy spaces on unbounded Vilenkin groups

AbstractWe characterize atomic Hardy spaces on unbounded locally compact Vilenkin groups by means of a modified maximal function. The obtained Fourier multiplier theorem is more general than the corresponding results due to Kitada, Onneweer and Quek, Daly and Phillips that were proved under the boundedness assumption on the underlying group

Concepts of generalized bounded variation and the theory of Fourier series

The aim of this article is to unify a large part of present knowledge about the behaviour of Fourier series of functions of generalized bounded variation. The connections between various concepts are discussed, particular attention being payed to those due to Waterman and Chanturiya. Exploiting the existing interactions and utilizing the power of one or another approach to some typical questions of the Fourier theory, a number of previously unnoticed results are obtained in the course of this exposition