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Results in Mathematics

, Volume 59, Issue 3–4, pp 359–400 | Cite as

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

  • P. L. Butzer
  • P. J. S. G. Ferreira
  • G. Schmeisser
  • R. L. Stens
Article

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.

Mathematics Subject Classification (2010)

65B15 65D32 94A20 

Keywords

Euler–Maclaurin summation formula Abel–Plana summation formula Poisson summation formula Whittaker–Kotel’nikov–Shannon sampling theorem Approximate sampling formula Bandlimited signals Quadrature formulae 

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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • P. L. Butzer
    • 1
  • P. J. S. G. Ferreira
    • 2
  • G. Schmeisser
    • 3
  • R. L. Stens
    • 1
  1. 1.Lehrstuhl A für MathematikRWTH Aachen UniversityAachenGermany
  2. 2.IEETA/DETIUniversidade de AveiroAveiroPortugal
  3. 3.Department of MathematicsUniversity of Erlangen-NürnbergErlangenGermany

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