Convergence in Lp-norm on the complete product of finite noncommutative groups

Abstract

Several 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. This structure was generalized by Vilenkin in 1947 studying the complete product of arbitrary cyclic groups. In Vilenkin groups the order of the cyclic groups appeared in the product can be unbounded. The methods applied in the study of these cases differ significantly from the bounded cases and in many instances we obtain different results for the same question.

A natural generalization of the Vilenkin groups is the complete product of arbitrary groups, non necessarily commutative groups. In this case we use representation theory in order to obtain orthonormal systems, taking the finite product of the normalized coordinate functions of the continuous irreducible representations appeared in the dual object of the finite groups. These systems are named representative product systems. Representative product systems can be represented on the interval $[0,1]$, where this systems are also orthonormal under the Lebesgue measure.

In this talk I deal with the convergence in $L^p$-norm of Fourier series, Fej\'er means and Ces\`aro means of order $\alpha$ with respect to representative product systems, where $1\le p<\infty$. We found an $0\le\alpha_0<\frac12$ such that the Ces\`aro means of order $\alpha$ of the Fourier series with respect to representative product systems on bounded groups converge to the function in $L^p$-norm ($1\le p<\infty$) for all $\alpha_0<\alpha<1$. On the other hand, we obtain an $0\le\alpha_1\le\alpha_0$ such that for all $0<\alpha<\alpha_1$ there exists an $f\in L^1(G)$ for which $\sigma^{\alpha}_nf$ does not converge to the function $f$ in $L^1$-norm.