On the converge in Lp-norm of Cesàro means with respect to representative product systems

Abstract

A natural generalization of the Vilenkin group is the complete product of arbitrary groups, which are non necessarily commutative. In this case we use representation theory in order to obtain the orthonormal systems with which we work. This systems are named representative product systems and they can be represented on the interval $[0,1]$, where the systems are also orthonormal using the Lebesgue measure. The author proved that the Fejér means of the Fourier series with respect to representative product systems converge to the function in $L^p$-norm ($1\le p<\infty$), although we can not state the same for the Fourier series in general.

In this talk I extend this statement for Cesàro means of order $\alpha$ where $0<\alpha<1$. However, I can not prove it for all $0<\alpha<1$. I found an $\alpha_0<\frac12$ depending on the values of the representative product systems for which the following theorem holds.

Theorem. If $G$ is a bounded group, $\alpha_0<\alpha<1$ and $f\in L^p(G)$, $1\le p<\infty$, then $\sigma^{\alpha}_nf \rightarrow f$ in $L^p$-norm, where $\sigma^{\alpha}_nf$ is the Cesàro means of order $\alpha$ of the function $f$.