Universal series in ∩p>1ℓp

Abstract

In this paper an abstract condition is given yielding universal series defined by sequences a = {aj}∞ j=1 in ∩p>1ℓp but not in ℓ1. We obtain a unification of some known results related to approximation by translates of specific functions including the Riemann ζ-function, or a fundamental solution of a given elliptic operator in ℝν with constant coefficients or an approximate identity as, for example, the normal distribution. Another application gives universal trigonometric series in simultaneously with respect to all σ-finite Borel measures in ℝν. Stronger results are obtained by using universal Dirichlet series. © 2009 London Mathematical Society.