Shannon's sampling theorem, incongruent residue classes and Plancherel's theorem
Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 425-437

Sampling theory for multi-band signals is shown to have a logical structure similar to that of Fourier analysis.

On montre que la théorie de l'échantillonnage pour les signaux multi-canaux a une structure logique qui s'apparente à celle de l'analyse de Fourier.

@article{JTNB_2002__14_2_425_0,
     author = {Maurice M. Dodson},
     title = {Shannon's sampling theorem, incongruent residue classes and {Plancherel's} theorem},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {425--437},
     year = {2002},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {2},
     zbl = {02184592},
     mrnumber = {2040686},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_425_0/}
}
TY  - JOUR
AU  - Maurice M. Dodson
TI  - Shannon's sampling theorem, incongruent residue classes and Plancherel's theorem
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2002
SP  - 425
EP  - 437
VL  - 14
IS  - 2
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_425_0/
LA  - en
ID  - JTNB_2002__14_2_425_0
ER  - 
%0 Journal Article
%A Maurice M. Dodson
%T Shannon's sampling theorem, incongruent residue classes and Plancherel's theorem
%J Journal de théorie des nombres de Bordeaux
%D 2002
%P 425-437
%V 14
%N 2
%I Université Bordeaux I
%U https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_425_0/
%G en
%F JTNB_2002__14_2_425_0
Maurice M. Dodson. Shannon's sampling theorem, incongruent residue classes and Plancherel's theorem. Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 425-437. https://jtnb.centre-mersenne.org/item/JTNB_2002__14_2_425_0/

[1] M.G. Beaty, M.M. Dodson, Abstract harmonic analysis and the sampling theorem. In Sampling theory in Fourier and signal analysis:Advanced topics (eds. J. R. Higgins and R. L. Stens), Oxford University Press, 1999, 233-265.

[2] M.G. Beaty, M.M. Dodson, Shannon's sampling theorem, Plancherrel's theorem and spectral translates, preprint, University of York, 2002.

[3] M.G. Beaty, M.M. Dodson, Shannon's sampling theorem and abstract harmonic analysis, preprint, University of York, 2002.

[4] M.G. Beaty, M.M. Dodson, J.R. Higgins, Approximating Paley-Wiener functions by smoothed step functions. J. Approx. Theory 78 (1994), 433-445. | Zbl | MR

[5] M.G. Beaty, J.R. Higgins, Aliasing and Poisson summation in the sampling theory of Paley-Wiener spaces. J. Fourier Anal. Appl. 1 (1994), 67-85. | Zbl | MR

[6] P.L. Butzer, A survey of the Whittaker-Shannon sampling theorem and some of its extensions. J. Math. Res. Exposition 3 (1983), 185-212. | Zbl | MR

[7] P.L. Butzer, A. Gessinger, The approximate sampling theorem, Poisson's sum formula, a decomposition theorem for Parseval's equation and their interconnections. In The Heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. Rivlin, Ann. Numer. Math. 4 (1997), 143-160. | Zbl | MR

[8] P.L. Butzer, M. Hauss, R.L. Stens, The sampling theorem and its unique role in various branches of mathematics. Mitt. Math. Ges. Hamburg 12 (1991), 523-527. | Zbl | MR

[9] P.L. Butzer, J.R. Higg, R.L. Stens, Sampling theory of signal analysis. In: Development of mathematics 1950-2000, Birkhäuser, Basel, 2002, 193-234. | Zbl | MR

[10] P.L. Butzer, G. Schmeisser, R.L. Stens, An Introduction to Sampling Analysis. In: Nonuniform Sampling, Theory and Practice (ed. F. Marvasti), Kluwer Academic/Plenum Publishers, New York, 2000, Chapter 2, 17-121. | MR

[11] M.M. Dodson, A.M. Silva, Fourier analysis and the sampling theorem, Proc. Royal Irish Acad. 85A (1985), 81-108. | Zbl | MR

[12] M.M. Dodson, A.M. Silva, V. Soucek, A note on Whittaker's cardinal series in harmonic analysis. Proc. Edinb. Math. Soc. 29 (1986), 349-357. | Zbl | MR

[13] S. Goldman, Information Theory. Dover, New York, 1968.

[14] J.R. Higgins, Five short stories about the cardinal series, Bull. Amer. Math. Soc. 12 (1985), 45-89. | Zbl | MR

[15] J.R. Higgins, Sampling theory in Fourier and signal analysis: Foundations. Clarendon Press, Oxford, 1996. | Zbl

[16] J.R. Higgins, G. Schmeisser, J.J. Voss, The sampling theorem and several equivalent results in analysis. J. Comput. Anal. Appl. 2 (2000) 333-371. | Zbl | MR

[17] A.J. Jerri, The Shannon sampling theorem - its various extensions and applications: a tutorial review. Proc. IEEE 65 (1977), 1565-1596. | Zbl

[18] I. Kluvánek, Sampling theorem in abstract harmonic analysis. Mat.-Fyz. Gasopis Sloven. Akad. Vied. 15 (1965), 43-48. | Zbl | MR

[19] K.S. Krishnan, A simple result in quadrature. Nature 162 (1948), 215. | Zbl | MR

[20] S.P. Lloyd, A sampling theorem for stationary (wide sense) stochastic processes. Trans. Amer. Math. Soc. 92 (1959), 1-12. | Zbl | MR

[21] R. J. MARKS II (ed.), Advanced topics in Shannon sampling and interpolation theory. Springer-Verlag, New York, 1993. | Zbl | MR

[22] Q.I. Rahman, G. Schmeisser, The summation formulae of Poisson, Plana, Euler-Maclaurin and their relationship. J. Math. Sci. 28 (1994), 151-171. | Zbl

[23] C.E. Shannon, A mathematical theory of communication. Bell System Tech. J. 27 (1948), 379-423, 623-656. | Zbl | MR

[24] C.E. Shannon, Communication in the presence of noise. Proc. IRE 37 (1949), 10-21. | MR

[25] E.T. Whittaker, On the functions which are represented by the expansions of the interpolation theory. Proc. Roy. Soc. Edinburgh, 35 (1915), 181-194. | JFM