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Spaceability for sets of bandlimited input functions and stable linear time-invariant systems with finite time blowup behavior

  • Methods of Signal Processing
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Abstract

The approximation of linear time-invariant systems by sampling series is studied for bandlimited input functions in the Paley–Wiener space PW 1π , i.e., bandlimited signals with absolutely integrable Fourier transform. It has been known that there exist functions and systems such that the approximation process diverges. In this paper we identify a signal set and a system set with divergence, i.e., a finite time blowup of the Shannon sampling expression. We analyze the structure of these sets and prove that they are jointly spaceable, i.e., each of them contains an infinite-dimensional closed subspace such that for any function and system pair from these subspaces, except for the zero elements, we have divergence.

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Authors and Affiliations

  1. Technische Universität München, Lehrstuhl für Theoretische Informationstechnik, Munich, Germany

    H. Boche & U. J. Mönich

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  1. H. Boche
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  2. U. J. Mönich
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Correspondence to H. Boche.

Additional information

Original Russian Text © H. Boche, U.J. Mönich, 2017, published in Problemy Peredachi Informatsii, 2017, Vol. 53, No. 2, pp. 70–90.

Supported by the Gottfried Wilhelm Leibniz Programme of the German Research Foundation (DFG).

Parts of this work were presented at the Workshop on Harmonic Analysis, Graphs and Learning at the Hausdorff Research Institute for Mathematics, Bonn, Germany, and at the 2016 European Signal Processing Conference [1].

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Boche, H., Mönich, U.J. Spaceability for sets of bandlimited input functions and stable linear time-invariant systems with finite time blowup behavior. Probl Inf Transm 53, 164–182 (2017). https://doi.org/10.1134/S0032946017020053

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