Fourier analysis on the complete product of noncommutative finite groups

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 a 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.

This work relates the properties of representative product systems emphasizing the difference between abelian and noncommutative groups. It also deals 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 $1\le p<\infty$.