Representative product systems

Abstract. In this paper we summarize the most relevant results with respect to the convergence of Fourier series based on representative product systems. We focus on monomial systems defined on the complete product of quaternion groups. The results clearly indicate that these systems can be considered as a natural extension of the Vilenkin systems to the non-abelian harmonic analysis.

Key words and phrases. Walsh functions, Vilenkin systems, representative product systems, characters, Fourier series, Fejér means, Cesàro means of order \(\alpha\), convergence in \(L^p\)-norm and almost everywhere, absolute convergence, quaternion groups.