On the regularized Whittaker-Kotel'nikov-Shannon sampling formula

Author:
Liwen Qian

Journal:
Proc. Amer. Math. Soc. **131** (2003), 1169-1176

MSC (2000):
Primary 41A80, 41A30; Secondary 65D25, 65G99, 94A24

DOI:
https://doi.org/10.1090/S0002-9939-02-06887-9

Published electronically:
October 24, 2002

MathSciNet review:
1948108

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Abstract | References | Similar Articles | Additional Information

Abstract: Error estimation is given for the regularized Whittaker-Kotel'nikov-Shannon (WKS) sampling formula, which was found to be accurate and robust for numerically solving partial differential equations. The result improves the convergence rate of existing results.

**1.**E. T. Whittaker,*On the functions which are represented by the expansion of the interpolation theory*, Proc. Roy. Soc. Edinburgh, sec. A,**35**(1915), 181-194.**2.**V. Kotel'nikov,*On the carrying capacity of the ``ether" and wire in telecommunications, material for the first All-Union Conference on Questions of Communications*, Izd. Red. Upr. Svyazi RKKA, Moscow, Russian (1933).**3.**C. E. Shannon,*A mathematical theory of communication*, Bell System Tech. J.**27**(1948), 379-423.**4.**Robert J. Marks II,*Introduction to Shannon Sampling and Interpolation Theory*, Spring-Verlag, 1991. MR**92j:41001****5.**A. I. Zayed,*Advances in Shannon's Sampling Theory*, CRC Press, 1993. MR**95f:94008****6.**J. R. Higgins,*Sampling theory in Fourier and signal analysis : foundations*, Oxford University Press, 1996.**7.**D. Jagerman,*Bounds for truncation error of the sampling expansion.*SIAM J. Appl. Math.,**14**(1966), 714-723. MR**35:4673****8.**P. L. Butzer and R. L. Stens,*A modification of the Whittaker-Kotel'nikov-Shannon sampling series,*Aequationés Mathematicae**28**(1985), 305-311. MR**87a:94016****9.**R. Gervais, Q. I. Rahman, and G. Schmeisser,*A bandlimited function simulating a duration-limited one*. Anniversary Volume on Approximation Theory and Functional Analysis, Birkhäuser, Basel, (1984), 355-362. MR**87i:41002****10.**G. W. Wei, D. S. Zhang, D. J. Kouri and D. K. Hoffman,*Lagrange Distributed Approximating Functionals*, Phys. Rev. Lett.**79**(1997), 775-779.**11.**G. W. Wei,*Quasi wavelets and quasi interpolating wavelets*, Chem. Phys. Lett.**296**(1998), 215-222.**12.**G. W. Wei,*Discrete singular convolution method for the Sine-Gordon equation*, Physica D**137**:3-4 (2000), 247-259. MR**2000j:65101****13.**G. W. Wei,*Solving quantum eigenvalue problems by discrete singular convolution*, J. Phys. B**33**(2000), 343-359.**14.**S. Guan, C.-H. Lai and G. W. Wei,*Fourier-Bessel Analysis of Patterns in a Circular Domain*, Physica D**151**:2-4 (2001), 83-98. MR**2002d:35178****15.**H. D. Pollak,*A Remark on `Elementary Inequalities for Mills' Ratio' by Yûsaku Komatu*, Rep. Stat. Appl. Res., JUSE**4**:3 (1956). MR**18:722a****16.**E. D. Rainville,*Special functions*, Macmillan, 1960. MR**21:6447**; reprint MR**52:14399**

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Additional Information

**Liwen Qian**

Affiliation:
Department of Computational Science, National University of Singapore, Singapore 117543

Address at time of publication:
Singapore–MIT Alliance (SMA), E4-4-10, National University of Singapore, Singapore 117576

Email:
qianlw@cz3.nus.edu.sg, smaqlw@nus.edu.sg

DOI:
https://doi.org/10.1090/S0002-9939-02-06887-9

Keywords:
Whittaker-Kotel'nikov-Shannon's sampling formula,
regularization,
error estimate

Received by editor(s):
April 23, 2001

Received by editor(s) in revised form:
November 8, 2001

Published electronically:
October 24, 2002

Communicated by:
David Sharp

Article copyright:
© Copyright 2002
American Mathematical Society