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 Probl. Peredachi Inf., 2015, Volume 51, Issue 3, Pages 41–69 (Mi ppi2179)

Methods of Signal Processing

Strong divergence for system approximations

H. Bochea, U. J. Mönichb

a Technische Universität München, Lehrstuhl für Theoretische Informationstechnik, Germany, Germany
b Massachusetts Institute of Technology, Research Laboratory of Electronics, New York, USA

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 $\mathcal{PW}_\pi^1$. 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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English version:
Problems of Information Transmission, 2015, 51:3, 240–266

Bibliographic databases:

UDC: 621.391.1+517

Citation: H. Boche, U. J. Mönich, “Strong divergence for system approximations”, Probl. Peredachi Inf., 51:3 (2015), 41–69; Problems Inform. Transmission, 51:3 (2015), 240–266

Citation in format AMSBIB
\Bibitem{BocMon15} \by H.~Boche, U.~J.~M\"onich \paper Strong divergence for system approximations \jour Probl. Peredachi Inf. \yr 2015 \vol 51 \issue 3 \pages 41--69 \mathnet{http://mi.mathnet.ru/ppi2179} \transl \jour Problems Inform. Transmission \yr 2015 \vol 51 \issue 3 \pages 240--266 \crossref{https://doi.org/10.1134/S0032946015030047} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000363254900004} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84944447229} 

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• http://mi.mathnet.ru/eng/ppi/v51/i3/p41

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. H. Boche, V. Pohl, “On the Strong Divergence of Hilbert Transform Approximations and a Problem of Ul'yanov”, J. Approx. Theory, 204 (2016), 34–60
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