Mean Convergence of Generalized Walsh-Fourier Series

Abstract

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.

Approximation by representative functions on the complete product of S3

“…This statement is known as the Paley's theorem. Paley's theorem was shown independently for arbitrary Vilenkin systems by Young [6], Schipp [7] and Simon [8]. Unfortunately, we can not extend this statement for the complete product of S 3 .…”

On the existence of universal series by the generalized Walsh system

“…The basic properties of the generalized Walsh system of order a were obtained by H. E. Chrestenson, R. Paley, J. Fine, W. Young, C. Watari, N. Vilenkin, and others (see [1], [8], [14], [18]- [20]). Next, we present some properties of the Ψ a system.…”

Boundedness of Sublinear Operators in Herz Spaces on Vilenkin Groups and its Application

“…In $2 of this paper, we will first establish the boundedness of some sublinear operators, which are bounded on L& (G) with 1 < q < 00, in the weighted Herz spaces on G under some weak hypotheses on the size of these operators. These hypotheses on the size are obviously local at the identity, and are satisfied by most of the important operators in harmonic analysis on Vilenkin groups, for example, the Hardy -Littlewood maximal operators, Calder6n -Zygmund operators, the nth partial sum operators of the Vilenkin-Fourier series, pseudo -differential operators and so on (see 111, [12], [14], [15], [19] and [20]). We then consider the similar questions for those sublinear operators which map P ( G ) to U ( G ) with 1 < p < q < 00.…”