Cardinal sine series, oversampling, and periodic distributions
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- by B. A. Bailey and W. R. Madych PDF
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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.References
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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
- 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
- © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3373935