Mean Convergence of Generalized Walsh-Fourier Series

Abstract

Paley proved that Walsh-Fourier series converges in I?(1 < p < ).We generalize Paley's result to Fourier series with respect to characters of countable direct products of finite cyclic groups of arbitrary orders.1. Introduction.It is known that the Walsh functions are characters of the countable direct product of groups of order 2. In this note we consider characters of IL10Z , where Zp. is a cyclic group of order p, p. > 2. Various Fourier properties of this generalized Walsh system have been studied in[2], and others.Many of these results are obtained only for the case where sup,-/?; < .In fact, Price [7] showed that some basic properties no longer hold when supp{ = .We will show that results concerning mean convergence, however, are still valid even if the orders p are unbounded.The bounded case was first obtained by Watari [9].See also Gosselin [2].The author would like to thank Richard Hunt and Charles Fefferman for several helpful conversations.Let {p,}I>o De a sequence of integers, p >2.Let G = IL~-0Zp.be the direct product of cyclic groups of order p, and p the Haar measure on G normalized by p(