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Divergence Behavior of Sequences of Linear Operators with Applications

  • Holger Boche
  • Ullrich J. Mönich
Article
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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., \(\ell ^p\), \(1\le p < \infty \), C[0, 1], \(L^p\), \(1< p <\infty \), 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.

Keywords

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

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Technische Universität München, Lehrstuhl für Theoretische InformationstechnikMünchenGermany

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