Negative results concerning Fourier series on the complete product of S3

Abstract

The aim of this paper is to continue the studies about convergence in $L^p$-norm of the Fourier series based on representative product systems on the complete product of finite groups. We restrict our attention to bounded groups with unbounded sequence $\Psi$. The most simple example of this groups is the complete product of $\mathcal{S}_3$. In this case we proved the existence of an $1<p<2$ number for which exists an $f\in L^p$ such that its n-th partial sum of Fourier series $S_n$ do not converge to the function $f$ in $L^p$-norm (see \cite{GTa}). In this paper we extend this ''negative'' result for all $1<p<\infty$ and $p\ne2$ numbers.