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Strong divergence for system approximations

  • Methods of Signal Processing
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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 PW 1π . 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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Authors and Affiliations

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

    H. Boche

  2. Massachusetts Institute of Technology, Research Laboratory of Electronics, New York, USA

    U. J. Mönich

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  1. H. Boche
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Correspondence to H. Boche.

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Original Russian Text © H. Boche, U.J. Mönich, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, No. 3, pp. 41–69.

The material in this paper was presented in part at the 2015 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’2015).

Partly supported by the German Research Foundation (DFG) under grant BO 1734/22-1.

Supported by the German Research Foundation (DFG) under grant MO 2572/1-1.

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Boche, H., Mönich, U.J. Strong divergence for system approximations. Probl Inf Transm 51, 240–266 (2015). https://doi.org/10.1134/S0032946015030047

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