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Journal of Fourier Analysis and Applications

, Volume 24, Issue 1, pp 285–308 | Cite as

System Representations for the Paley–Wiener Space \(\mathcal {PW}_{\pi }^2\)

  • Holger Boche
  • Ullrich J. Mönich
Article

Abstract

In this paper we study the approximation of stable linear time-invariant systems for the Paley–Wiener space \(\mathcal {PW}_{\pi }^2\), i.e., the set of bandlimited functions with finite \(L^2\)-norm, by convolution sums. It is possible to use either, the convolution sum where the time variable is in the argument of the bandlimited impulse response, or the convolution sum where the time variable is in the argument of the function, as an approximation process. In addition to the pointwise and uniform convergence behavior, the convergence behavior in the norm of the considered function space, i.e. the \(L^2\)-norm in our case, is important. While it is well-known that both convolution sums converge uniformly on the whole real axis, the \(L^2\)-norm of the second convolution sum can be divergent for certain functions and systems. We show that the there exist an infinite dimensional closed subspace of functions and an infinite dimensional closed subspace of systems, such that for any pair of function and system from these two sets, we have norm divergence.

Keywords

Paley–Wiener space Bandlimited function Linear time-invariant system Convolution sum Divergence \(L^2\)-norm 

Mathematics Subject Classification

94A12 94A20 15A03 

Notes

Acknowledgements

This work was supported by the Gottfried Wilhelm Leibniz Programme of the German Research Foundation (DFG) under Grant BO 1734/20-1, and U. Mönich was supported by the German Research Foundation (DFG) under Grant MO 2572/2-1. It was motivated by questions raised by Hans Feichtinger about sampling-type system approximation processes and the structure of signal sets creating divergence. Parts of this work were presented at the Workshop on Harmonic Analysis, Graphs and Learning at the Hausdorff Research Institute for Mathematics, Bonn, Germany. Holger Boche would like to thank Hans Feichtinger for fruitful discussions and the Hausdorff Research Institute for Mathematics for its support and hospitality.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Lehrstuhl für Theoretische InformationstechnikTechnische Universität MünchenMunichGermany

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