Completeness of an exponential system in weighted Banach spaces and closure of its linear span

E. Zikkos

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

For a real multiplicity sequence Λ={ λn, μn }n = 1, that is, a sequence where { λn } are distinct positive real numbers satisfying 0 < λn < λn + 1 {mapping} ∞ as n {mapping} ∞ and where each λn appears μn times, we associate the exponential systemEΛ = { tk eλn t : k = 0, 1, 2, ..., μn - 1 }n = 1 .For a certain class of multiplicity sequences, we give necessary and sufficient conditions in order for EΛ to be complete in some weighted Banach space of continuous functions on R, and in some weighted Lp (- ∞, ∞) spaces of measurable functions, with p ∈ [1, ∞). We also prove that if EΛ is incomplete in the weighted spaces, then every function in the closure of the linear span of EΛ*, where EΛ * = { tμn - 1 eλn t }n = 1, can be extended to an entire function represented by a Taylor-Dirichlet seriesg (z) = underover(∑, n = 1, ∞) cn zμn - 1 eλn z, cn ∈ C .Furthermore, we prove that EΛ is minimal in the weighted spaces if and only if it is incomplete.

Original languageBritish English
Pages (from-to)115-148
Number of pages34
JournalJournal of Approximation Theory
Volume146
Issue number1
DOIs
StatePublished - May 2007

Keywords

  • Closure
  • Completeness
  • Minimality
  • Taylor-Dirichlet series

Fingerprint

Dive into the research topics of 'Completeness of an exponential system in weighted Banach spaces and closure of its linear span'. Together they form a unique fingerprint.

Cite this