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The Summation Formulae of Euler–Maclaurin, Abel–Plana, Poisson, and their Interconnections with the Approximate Sampling Formula of Signal Analysis

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Abstract

This paper is concerned with the two summation formulae of Euler–Maclaurin (EMSF) and Abel–Plana (APSF) of numerical analysis, that of Poisson (PSF) of Fourier analysis, and the approximate sampling formula (ASF) of signal analysis. It is shown that these four fundamental propositions are all equivalent, in the sense that each is a corollary of any of the others. For this purpose ten of the twelve possible implications are established. Four of these, namely the implications of the grouping \({\text{APSF}\Leftarrow\text{ASF}\Rightarrow\text{EMSF}\Leftrightarrow\text{PSF}}\) are shown here for the first time. The proofs of the others, which are already known and were established by three of the above authors, have been adapted to the present setting. In this unified exposition the use of powerful methods of proof has been avoided as far as possible, in order that the implications may stand in a clear light and not be overwhelmed by external factors. Finally, the four propositions of this paper are brought into connection with four propositions of mathematical analysis for bandlimited functions, including the Whittaker–Kotel’nikov–Shannon sampling theorem. In conclusion, all eight propositions are equivalent to another. Finally, the first three summation formulae are interpreted as quadrature formulae.

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Authors and Affiliations

  1. Lehrstuhl A für Mathematik, RWTH Aachen University, 52056, Aachen, Germany

    P. L. Butzer & R. L. Stens

  2. IEETA/DETI, Universidade de Aveiro, 3810-193, Aveiro, Portugal

    P. J. S. G. Ferreira

  3. Department of Mathematics, University of Erlangen-Nürnberg, Bismarckstraße 1 1/2, 91054, Erlangen, Germany

    G. Schmeisser

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  1. P. L. Butzer
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  2. P. J. S. G. Ferreira
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Correspondence to P. L. Butzer.

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In Memory of Alexander Ostrowski, 1893–1986

Dedicated to Heinrich Wefelscheid to mark his 70th birthday

After this manuscript was accepted for publication the authors learned that special issues were being prepared to honour Heinrich Wefelscheid on the occasion of his 70th anniversary. Since they valued his unstinted service to mathematics as a whole, thus the founding of the present journal and his leading editorship over the years, his publication of the Collected Works, in several volumes, of W. Blaschke and E. Landau, they expressed interest that the present paper, already dedicated to A. Ostrowski, be included in the special issues.

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Butzer, P.L., Ferreira, P.J.S.G., Schmeisser, G. et al. The Summation Formulae of Euler–Maclaurin, Abel–Plana, Poisson, and their Interconnections with the Approximate Sampling Formula of Signal Analysis. Results. Math. 59, 359–400 (2011). https://doi.org/10.1007/s00025-010-0083-8

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