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The Whittaker-Kotel'nikov-Shannon theorem, spectral translates and Plancherel's formula

Beaty, M.G. and Dodson, M.M. (2004) The Whittaker-Kotel'nikov-Shannon theorem, spectral translates and Plancherel's formula. Journal of Fourier Analysis and Applications, 10 (2). pp. 179-199. ISSN 1531-5851

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Abstract

The classical sampling or WKS theorem on reconstructing signals from uniformly spaced samples assumes the signals to be band-limited (i.e., with spectrum in a bounded interval [-W,W]). This assumption was later weakened to a disjoint translates condition on the spectrum which led to an extension of the sampling theorem to multi-band signals with spectra in a union of finite intervals. In this article the disjoint translates condition is replaced by a more natural lsquonull intersectionrsquo condition on spectral translates. This condition is shown to be equivalent to an analogue of Plancherelrsquos isometric formula when the spectrum has finite measure. Thus to some extent multi-band sampling theory has a logical structure similar to classical Fourier analysis. The relationship between the null intersection condition and the isometric formula is illustrated by considering the consequences when the null intersection condition does not hold. In this case, the sampling representation cannot hold for any function, whereas the isometric formula can still hold for some functions.

Item Type: Article
Institution: The University of York
Academic Units: The University of York > Mathematics (York)
Depositing User: York RAE Import
Date Deposited: 22 Apr 2009 08:47
Last Modified: 22 Apr 2009 08:47
Published Version: http://dx.doi.org/10.1007/s00041-004-8010-6
Status: Published
Publisher: Springer Science+Business Media
Identification Number: 10.1007/s00041-004-8010-6
URI: http://eprints.whiterose.ac.uk/id/eprint/6687

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