Universal transforms of the geometric series under generalized Riesz methods

Abstract

In this paper generalized Riesz methods (R, p, M) of summability are considered. We prove that, to each open set O ⊂ C with adequate topological properties and each sequence {Pn} ⊂ C tending to infinity, we can associate a corresponding P-regular (R, p, M)-method so that the geometric series and a certain trigonometric series become universal in the sense that its (R, p, M)-transforms approximate any member of certain spaces of holomorphic functions or measurable functions.