Fourier analysis on the complete product of finite groups

Abstract

We present results about the Fourier analysis on the complete product of arbitrary finite 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 epresentations. 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 $0\le j<k$.

For the commutative case the $n$th partial sum of Fourier series are bounded, uniformly in $n$, from $L^p$ into itself for $1<p<\infty$. This statement is only true for $p=2$ if we take the complete product of non-abelian finite groups with bounded order but unbounded sequence $\Psi$. However, the Fej{\'e}r means of Fourier series converge in $L^p$-norm ($p\ge 1$) to the function when the order of all of the finite groups is bounded.

On the other hand, we consider an nteresting class of functions, namely the ones that are constant on every conjugacy classes. The system of characters of the representations is complete in this class of functions, so we use characters to study the convergence of series constructed in this way.