On the convergence in L 1-norm of Cesàro means with respect to representative product systems

Abstract

The aim of this paper is to prove that the Cesàro means of order α (0 < α < 1) of the Fourier series with respect to representative product systems converge to the function in L 1 -norm, only for certain values of α which depend on some parameter of the representative product system. IntroductionSeveral results in Fourier analysis with respect to Walsh functions are obtained viewing them as the characters of the dyadic group, i.e., the complete product of the discrete cyclic group of order 2 with the product of topologies and measures. Then we often order the Walsh functions in Paley's sense writing them as a nite product of the Rademacher functions. It is named the WalshPaley system. The above structure was generalized by Vilenkin [7] in 1947 studying the complete product of arbitrary cyclic groups. The construction of the system here is similar, taking the nite product of the characters of the cyclic groups as it Paley did.A natural generalization of the Vilenkin group is the complete product of arbitrary, not necessarily commutative groups. In this case we use representation theory in order to obtain orthonormal systems. These systems are named representative product systems and Section 1 deals with their denition and properties. In [2] the authors proved the fact that the Fejér means of the Fourier series with respect to representative product systems on bounded groups converge to the function in L p -norm (1 p < ∞), although we can not state the same for the Fourier series in general. We attempt to extend this statement to Cesàro means of order α where 0 < α < 1.In Section 2 we dene the basic concept and properties concerning Cesàro means and prove the lemmas we need for our main result in Section 3: There is an 0 α 0 < 1 2 such that the Cesàro means of order α of the Fourier series with respect to representative product systems on bounded groups converge to the function in L p -norm (1 p < ∞) for all α 0 < α < 1. This result was proved by Skvortsov (see [5]) for the commutative cases and for all 0 < α < 1. Corollary 1 generalizes this result for monomial representative product systems. In Section 3 we also deal with the divergence of the Cesàro means of order α and we obtain an 0 α 1 α 0 such that for all 0 < α < α 1 there exists an f ∈ L 1 (G) for which σ α n f does not converge to the function f in L 1 -norm.Throughout this work we denote by N, P, C the set of nonnegative, positive integers and complex numbers, respectively. In order to simplify notation we always use multiplication to denote the group operation and use the symbol e to denote the identity of the groups. If in our notations the indexset of a sum or a product is empty then the value of the sum and the product is 0 and 1 respectively. Moreover throughout this paper we often use constants C which have dierent values in dierent occurrences and they only depend on the value of α. The notation which we used in this paper is similar to the one in [3]. Representative product systemsLet m := (m k , k ∈ N) be a sequence of positive integers such tha...

Approximation by representative functions on the complete product of S3

“…For values of α less than log 6 4 3 we obtain divergence in L 1 -norm. Theorem 7 (see [11]). Let G be the complete product of S 3 and ψ be the representative product system with respect to the system ϕ of Table 1.…”