Shannon's sampling theorem, incongruent residue classes and Plancherel's theorem
Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, p. 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 = {Dodson, Maurice M.},
     title = {Shannon's sampling theorem, incongruent residue classes and Plancherel's theorem},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {2},
     year = {2002},
     pages = {425-437},
     zbl = {02184592},
     mrnumber = {2040686},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2002__14_2_425_0}
}
Dodson, Maurice M. Shannon's sampling theorem, incongruent residue classes and Plancherel's theorem. Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 425-437. http://www.numdam.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. | MR 1292971 | Zbl 0807.41011

[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. | MR 1307069 | Zbl 0839.94002

[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. | MR 724869 | Zbl 0523.94003

[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. | MR 1422676 | Zbl 0928.42022

[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. | MR 1144805 | Zbl 0824.94007

[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. | MR 1796842 | Zbl 0961.94009

[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 1875676

[11] M.M. Dodson, A.M. Silva, Fourier analysis and the sampling theorem, Proc. Royal Irish Acad. 85A (1985), 81-108. | MR 821425 | Zbl 0583.42003

[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. | MR 865268 | Zbl 0616.43006

[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. | MR 766960 | Zbl 0562.42002

[15] J.R. Higgins, Sampling theory in Fourier and signal analysis: Foundations. Clarendon Press, Oxford, 1996. | Zbl 0872.94010

[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. | MR 1793189 | Zbl 1030.41010

[17] A.J. Jerri, The Shannon sampling theorem - its various extensions and applications: a tutorial review. Proc. IEEE 65 (1977), 1565-1596. | Zbl 0442.94002

[18] I. Kluvánek, Sampling theorem in abstract harmonic analysis. Mat.-Fyz. Gasopis Sloven. Akad. Vied. 15 (1965), 43-48. | MR 188717 | Zbl 0154.44403

[19] K.S. Krishnan, A simple result in quadrature. Nature 162 (1948), 215. | MR 25613 | Zbl 0030.16301

[20] S.P. Lloyd, A sampling theorem for stationary (wide sense) stochastic processes. Trans. Amer. Math. Soc. 92 (1959), 1-12. | MR 107301 | Zbl 0106.11902

[21] R. J. MARKS II (ed.), Advanced topics in Shannon sampling and interpolation theory. Springer-Verlag, New York, 1993. | MR 1221743 | Zbl 0905.94002

[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 1019.65500

[23] C.E. Shannon, A mathematical theory of communication. Bell System Tech. J. 27 (1948), 379-423, 623-656. | MR 26286 | Zbl 1154.94303

[24] C.E. Shannon, Communication in the presence of noise. Proc. IRE 37 (1949), 10-21. | MR 28549

[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 45.1275.02