Recent developments on approximation theory with respect to representative product system

Abstract

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 and they are orthonormal and complete systems in L1 , but not necessary uniformly bounded. Representative product systems can be represented on the interval [0, 1], where this systems are also orthonormal under the Lebesgue measure. This work relates the recent results I obtained in this topic. It also deals with the differences between a Vilenkin system and a general representative product system based in the comportment of Dirichlet kernels.