A generalization of Walsh and Vilenkin systems

Abstract

I deal with Fourier analysis on the complete product of arbitrary finite groups starting from the results obtained in the commutative case. The Walsh-Paley system is defined on the complete product of the cyclic group of order 2 and Vilenkin systems are defined on the complete product of arbitrary cyclic groups. In order to define an orthonormal system in the not necessarily commutative case we follow the way given by harmonic analysis. Take the continuous irreducible representations appeared in the dual object of the finite groups. Let $\psi$ be the product system of the normalized coordinate functions of this representations. Thus we say that $\psi$ is a representative product system. This system is orthonormal and complete in ${L}^1$, but not necessary uniformly bounded, so define the sequence $\Psi_k$ by the multiplication of the greatest product of the $L^{\infty}$-norm and the $L^1$-norm of the coordinate functions appeared in all finite group $G_j$ for $0\le j<k$. I study the convergence in $L^p$-norm ($p\ge 1$) of the Fourier series and Fej{\'e}r means with respect to representative product systems.