Problems of Information Transmission

, Volume 51, Issue 3, pp 240–266 | Cite as

Strong divergence for system approximations

Methods of Signal Processing

Abstract

In this paper we analyze approximation of stable linear time-invariant systems, like the Hilbert transform, by sampling series for bandlimited functions in the Paley–Wiener space PWπ1. It is known that there exist systems and functions such that the approximation process is weakly divergent, i.e., divergent for certain subsequences. Here we strengthen this result by proving strong divergence, i.e., divergence for all subsequences. Further, in case of divergence, we give the divergence speed. We consider sampling at Nyquist rate as well as oversampling with adaptive choice of the kernel. Finally, connections between strong divergence and the Banach–Steinhaus theorem, which is not powerful enough to prove strong divergence, are discussed.

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Copyright information

© Pleiades Publishing, Inc. 2015

Authors and Affiliations

  1. 1.Technische Universität MünchenLehrstuhl für Theoretische InformationstechnikGermanyGermany
  2. 2.Massachusetts Institute of TechnologyResearch Laboratory of ElectronicsNew YorkUSA

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