On the approximate form of Kluvánek's theorem

Authors:
M. G. Beaty;M. M. Dodson;S. P. Eveson;J. R. Higgins
Affiliations:
Department of Mathematics, University of Newcastle, Newcastle NE1 7RU, UK;Department of Mathematics, University of York, York YO10 5DD, UK;Department of Mathematics, University of York, York YO10 5DD, UK;IHP, 4 rue du Bary, 11250 Montclar, Carcassonne, France
Venue:
Journal of Approximation Theory
Year:
2009

Citing 3
Cited 0

Approximation of continuous and discontinuous functions by generalized sampling series

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A generalized sampling theorem for locally compact abelian groups

Mathematics of Computation
Classical and approximate sampling theorems: studies in the LP (R) and the uniform norm

Journal of Approximation Theory

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Abstract

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 Kluvanek's theorem is established.