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Shannon Sampling Theorem via Poisson, Cauchy, Jacobi, and Levin

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Sampling, Approximation, and Signal Analysis

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Notes

  1. 1.

    Young provides an excellent discussion of this material in [75].

  2. 2.

    Please also see [31, 34, 36, 55, 56, 58, 59, 66, 69, 73, 74, 76].

  3. 3.

    We say the \(\psi (r) <^{\mathrm {as}} \phi (r)\) if \(\psi (r) < \phi (r)\) for all r sufficiently large.

  4. 4.

    Chapter 3 of Prenter [64] provides an excellent discussion of these ideas.

  5. 5.

    Please also see [7, 8, 10,11,12, 15, 19, 20, 24, 25, 70].

  6. 6.

    It is of interest to compare the envelope condition above with the Paley-Wiener-Schwartz growth condition. Note that Hörmander’s envelope condition is essentially the inversion of the Paley-Wiener-Schwartz growth condition.

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Acknowledgements

The author thanks Professors J. Rowland Higgins, Carlos Berenstein, John Benedetto, Radu Balan, and David Walnut for several conversations related to the contents of the chapter. Professor Higgins’ passing was a sad event for our community. The author wishes to acknowledge his deep respect for J. R. Higgins. He was a truly excellent mathematician, an extremely knowledgeable math historian, and an even better person. He was a true blessing to our community.

The author’s research was partially supported by US Air Force Office of Scientific Research Grant Number FA9550-20-1-0030.

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  1. Department of Mathematics and Statistics, American University, Washington, DC, USA

    Stephen D. Casey

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  1. Department of Mathematics and Statistics, American University, Washington, DC, USA

    Stephen D. Casey

  2. Department of Mathematics, University of York, York, UK

    M. Maurice Dodson

  3. IEETA, University of Aveiro, Aveiro, Portugal

    Paulo J. S. G. Ferreira

  4. Department of Mathematical Sciences, DePaul University, Chicago, IL, USA

    Ahmed Zayed

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Casey, S.D. (2023). Shannon Sampling Theorem via Poisson, Cauchy, Jacobi, and Levin. In: Casey, S.D., Dodson, M.M., Ferreira, P.J.S.G., Zayed, A. (eds) Sampling, Approximation, and Signal Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-41130-4_8

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