The maximal value of Dirichlet kernels with respect to representative product systems

Abstract

The Walsh-Paley system is formed by the characters of the dyadic group, i.e., the complete product of the discrete cyclic group of order 2 with the product of topologies and measures. Vilenkin in 1947 generalized this structure studying the complete product of arbitrary cyclic groups. In Vilenkin groups the order of the cyclic groups appeared in the product can be unbounded. A natural generalization of the Vilenkin groups is the complete product of arbitrary groups, non necessarily commutative groups. In this case we use representation theory in order to obtain orthonormal systems, taking the finite product of the normalized coordinate functions of the continuous irreducible representations appeared in the dual object of the finite groups. These systems are named representative product systems. In this talk I deal with the differences between a Vilenkin system and a general representative product system based in the comportment of Dirichlet kernels.