Divergence Behavior of Sequences of Linear Operators with Applications

Holger Boche, Ullrich J. Mönich

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we study the spaceability of divergence sets of sequences of bounded linear operators on Banach spaces. For Banach spaces with the s-property, we can give a sufficient condition that guarantees the unbounded divergence on a set that contains an infinite dimensional closed subspace after the zero element has been added. This generalizes the classical Banach–Steinhaus theorem which implies that the divergence set is a residual set. We further prove that many important spaces, e.g., ℓ p , 1 ≤ p< ∞, C[0, 1], L p , 1 < p< ∞, as well as Paley–Wiener and Bernstein spaces, have the s-property. Finally, consequences for the convergence behavior of sampling series and system approximation processes are shown.

Original languageEnglish
Pages (from-to)427-459
Number of pages33
JournalJournal of Fourier Analysis and Applications
Volume25
Issue number2
DOIs
StatePublished - 15 Apr 2019

Keywords

  • Banach–Steinhaus theorem
  • Paley–Wiener spaces
  • Sampling series
  • Spaceability
  • System and signal approximation

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