The zero set Λ of exponential polynomials and the Riesz property of its exponential system EΛ in L2(0 , T)

Elias Zikkos

Research output: Contribution to journalArticlepeer-review

Abstract

Consider the class of exponential polynomials of the form f(z)=∑n=0N(∑k=0mncn,kzk)ehnz,0=h0<h1<h2<⋯<hN,such that the coefficients cn,k are complex numbers with cn,mn≠0 for all n∈ { 0 , 1 , … , N}. Let Λf={λn,μn}n∈Z be the zero set of f, where {λn,μn}n∈Z:={…,λ-1,…,λ-1⏟μ-1-times,λ0,…,λ0⏟μ0-times,λ1,…,λ1⏟μ1-times,…}.We show that the upper Beurling uniform density of Λ f is equal to hN/ (2 π). Then based on results by Avdonin and Moran, we prove that if Λ f is of neutral type, then there is a family of functions of generalized divided differences, denoted by E(Λ) , belonging to the span of the exponential system EΛ={tkeλnt:n∈Z,k=0,1,…,μn-1}, such that E(Λ) is a Riesz sequence in L2(0 , T) for any T> hN. If Λ f is not of neutral type, we show that there exist real numbers un satisfying | Re (λn+ un) | = O(1) , so that the exponential system {tke(λn+un)t:n∈Z,k=0,1,…,μn-1}is a Riesz sequence in L2(0 , T) for any T> hN.

Original languageBritish English
Pages (from-to)307-319
Number of pages13
JournalArchiv der Mathematik
Volume120
Issue number3
DOIs
StatePublished - Mar 2023

Keywords

  • Beurling densities
  • Divided differences
  • Exponential polynomials
  • Neutral type
  • Riesz sequences

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