Strong divergence for system approximations

Strong divergence for system approximations 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 π 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

Strong divergence for system approximations

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Publisher
Pleiades Publishing
Copyright
Copyright © 2015 by Pleiades Publishing, Inc.
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946015030047
Publisher site
See Article on Publisher Site

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 π 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.

Journal

Problems of Information TransmissionSpringer Journals

Published: Oct 18, 2015

References

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