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Cardinal sine series, oversampling, and periodic distributions


Authors: B. A. Bailey and W. R. Madych
Journal: Proc. Amer. Math. Soc. 143 (2015), 4373-4382
MSC (2010): Primary 30D10, 40A30, 94A20
DOI: https://doi.org/10.1090/S0002-9939-2015-12585-3
Published electronically: March 24, 2015
MathSciNet review: 3373935
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Abstract: Suppose $ u(n/\rho )$, $ n=0, \pm 1, \pm 2, \ldots $, are samples of a frequency band limited function $ u(z)$ and $ \rho $ is greater than the Nyquist rate. If the even part of these samples, $ u_e(n/\rho )$, has less than quadratic growth and the odd part, $ u_o(n/\rho )$, has less than linear growth as $ n \to \pm \infty $, then we show that the corresponding cardinal sine series, that plays an essential role in the Whittaker-Kotelnikov-Shannon sampling theorem, converges uniformly to $ u(z)$ on compact subsets of the complex plane. An appropriately adapted version of the technique used to prove this allows us to obtain a result concerning the local convergence of the Fourier series of periodic distributions.


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Additional Information

B. A. Bailey
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: benjamin.bailey@uconn.edu

W. R. Madych
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: madych@math.uconn.edu

DOI: https://doi.org/10.1090/S0002-9939-2015-12585-3
Keywords: Cardinal sine series, entire functions of exponential type, oversampling, periodic distributions
Received by editor(s): March 8, 2014
Received by editor(s) in revised form: June 11, 2014, and June 18, 2014
Published electronically: March 24, 2015
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2015 American Mathematical Society