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

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

Content Provider	Springer Nature Link
Author	Stens, R. L. Butzer, P. L. Schmeisser, G. Ferreira, P. J. S. G.
Copyright Year	2010
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.
Ending Page	400
Page Count	42
Starting Page	359
File Format	PDF
ISSN	14226383
e-ISSN	14209012
Journal	Results in Mathematics
Issue Number	3
Volume Number	59
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2011-04-02
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Quadrature formulae Sampling theory Abel–Plana summation formula Approximate sampling formula Euler–Maclaurin summation formula Mathematics Euler-Maclaurin formula Poisson summation formula Quadrature and cubature formulas Whittaker–Kotel’nikov–Shannon sampling theorem Bandlimited signals
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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DOI 10.1007/s00025-010-0083-8 Results in Mathematics The Summation Formulae of Euler–Maclaurin, Abel–Plana, Poisson, and their Interconnections with the Approximate Sampling Formula of Signal Analysis (2011)

Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections

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Analog of the Whittaker-Kotel’nikov-Shannon theorem from the point of view of Fourier-Bessel analysis

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The sampling theorem, Poisson's summation formula, general Parseval formula, reproducing kernel formula and the Paley-Wiener theorem for bandlimited signals -- their interconnections (2010)

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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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