Universal approximation theorem for Dirichlet series

O Demanze,A Mouze

doi:10.1155/ijmms/2006/37014

Abstract

The paper deals with an extension theorem by Costakis and Vlachou on simultaneous approximation for holomorphic function to the setting of Dirichlet series, which are absolutely convergent in the right half of the complex plane. The derivation operator used in the analytic case is substituted by a weighted backward shift operator in the Dirichlet case. We show the similarities and extensions in comparing both results. Several density results are proved that finally lead to the main theorem on simultaneous approximation.

Highlights

Let f (s) = n≥1 ann−s be a Dirichlet series and let σa( f ) be its abscissa of absolute convergence, defined by σa( f ) = inf σ ∈ R; an n−σ converges . n≥1 (1.1)We denote n≥1 ann−s σ = n≥1 |an|n−σ ∈ [0, +∞] for all σ ∈ R
The paper deals with an extension theorem by Costakis and Vlachou on simultaneous approximation for holomorphic function to the setting of Dirichlet series, which are absolutely convergent in the right half of the complex plane
Several density results are proved that lead to the main theorem on simultaneous approximation

Summary

UNIVERSAL APPROXIMATION THEOREM FOR DIRICHLET SERIES

MOUZE Received 14 September 2005; Revised 11 May 2006; Accepted 30 May 2006. The paper deals with an extension theorem by Costakis and Vlachou on simultaneous approximation for holomorphic function to the setting of Dirichlet series, which are absolutely convergent in the right half of the complex plane. The derivation operator used in the analytic case is substituted by a weighted backward shift operator in the Dirichlet case. We show the similarities and extensions in comparing both results. Several density results are proved that lead to the main theorem on simultaneous approximation

Introduction

Fix ε

We deduce from above that P satisfies