Abstract
Cauchy’s integral formula is compared with Shannon’s sampling theorem. It is shown that each can be deduced from the other by elementary means.
The second named author was supported by the Stiftung Volkswagenwerk.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ahlfors, L.V., Complex Analysis (3rd Ed.). McGraw-Hill Book Company, New York, 1979.
Apostol, T.M., Mathematical Analysis. Addison-Wesley Publishing Company, Reading, 1957.
Behnke, H. — Sommer, F., Theorie der analytischen Funktionen einer komplexen Veränderlichen (Studienausgabe, 3rd Ed.).Springer Verlag, Berlin, 1976.
Butzer, P.L., A survey of the Whittaker -Shannon sampling theorem and some of its extensions. J.Math. Res. Exposition 3(1983), 185–212.
Butzer, P.L. — Nessel, R.J., Fourier Analysis and Approximation. Birkhäuser Verlag, Basel and Academic Press, New York, 1971.
Butzer, P.L. — Ries, S. — Stens, R.L., The Whittaker — Shannon sampling theorem, related theorems and extensions. In: Proc. Jordan — IEEE Conf. (Amman, 25.–28.4.1984).
Butzer, P.L. — Splettstosser, W., A sampling theorem for duration-limited functions with error estimates. Inform. and Control 34 (1977), 55–65.
Butzer, P.L. — Stens, R.L., The Poisson summation formula, Whittaker’s cardinal series and approximate Integration. In: Proc. Second Edmonton Conf. on Approximation Theory (Edmonton, 7.–11.6.1983; Eds. Z. Ditzian — A. Meir — S. Riemenschneider — A. Sharma). Amer. Math Soc, Providence, 1983; pp. 19–36.
Churkin, J.I. — Jakowlew, C.P. — Wunsch, G., Theorie und Anwendung der Signalabtastung. Verlag Technik, Berlin, 1966.
Fichtenholz, G.M., Differential- und Integralrechnung II (6th Ed.). VEB Deutscher Verlag der Wissenschaften, Berlin, 1974.
Hille, E., Analytic Function Theory I. Ginn and Company, Boston, 1959.
Jagerman, D.L. — Fogel, L., Some general aspects of the sampling theorem. IRE Trans. Inform. Theory IT-2, (1956), 139–146.
Knopp, K., Theory and Applications of Infinite Series. Hafner Publishing Company, New York, 1971.
Nikol’skií, S.M., Approximation of Functions of Several Variables and and Imbedding Theorems. Springer — Verlag, Berlin, 1975.
Ries, S. — Stens, R.L., Pointwise convergence of sampling series. In: Proc. Second European Signal Processing Conf. (Erlangen, 12. — 16.9.1983; Ed. H.W. Schüssler). North Holland Publishing Company, Amsterdam, 1983, pp. 5–7.
de la Vallée — Poussin, Ch.J., Sur la convergence des formules d’inter-polation entre ordonnées équidistantes. Bull. Acad. Roy. Belg. 1908, 319–410.
Wunsch, G., Systemtheorie der Informationstechnik. Akademische Verlags-gesellschaft Geest & Portig, Leipzig, 1971.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1984 Springer Basel AG
About this chapter
Cite this chapter
Butzer, P.L., Ries, S., Stens, R.L. (1984). Shannon’s Sampling Theorem Cauchy’s Integral Formula, and Related Results. In: Butzer, P.L., Stens, R.L., Sz.-Nagy, B. (eds) Anniversary Volume on Approximation Theory and Functional Analysis. ISNM 65: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 65. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5432-0_33
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5432-0_33
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5434-4
Online ISBN: 978-3-0348-5432-0
eBook Packages: Springer Book Archive