On the approximate form of Kluvánek's theorem

An abstract form of the classical approximate sampling theorem is proved for functions on a locally compact abelian group that are continuous, square-integrable and have integrable Fourier transforms. An additional hypothesis that the samples of the function are square-summable is needed to ensure t...

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Bibliographic Details
Published in:Journal of approximation theory 2009-09, Vol.160 (1), p.281-303
Main Authors: Beaty, M.G., Dodson, M.M., Eveson, S.P., Higgins, J.R.
Format: Article
Language:English
Subjects:
Abstract harmonic analysis
An approximate Kluvánek theorem
Sampling theory
Signal theory
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Summary:An abstract form of the classical approximate sampling theorem is proved for functions on a locally compact abelian group that are continuous, square-integrable and have integrable Fourier transforms. An additional hypothesis that the samples of the function are square-summable is needed to ensure the convergence of the sampling series. As well as establishing the representation of the function as a sampling series plus a remainder term, an asymptotic formula is obtained under mild additional restrictions on the group. In conclusion a converse to Kluvánek's theorem is established.
ISSN:0021-9045
1096-0430
DOI:10.1016/j.jat.2009.02.013