Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T16:05:40.504Z Has data issue: false hasContentIssue false
  • English
  • Français

The standard summation operator, the Euler-Maclaurin sum formula, and the Laplace transformation

Part of: Special classes of linear operators Acceleration of convergence

Published online by Cambridge University Press:  09 April 2009

John Boris Miller
John Boris Miller
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A proof is given of the Euler-Maclaurin sum formula, on a Banach space of differentiable vector-valued functions of bounded exponential growth, using the Laplace transformation. Some related summation formulae are proved by the same methods. Properties of the standard summation operator are proved, namely spectral properties and boundedness, continuity and differentiability results.

Keywords

standard summation operatorEuler-Maclaurin formulaLaplace transformationfunctions of exponential growthdifferentiation operatorintegration operatorBernoulli polynomialsperiodic extensionperiodic Bernoulli polynomialsfunctional equation

MSC classification

Secondary: 65B15: Euler-Maclaurin formula 47B38: Operators on function spaces (general)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 39 , Issue 3 , December 1985 , pp. 367 - 390
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Berndt, B. C. and Schoenfeld, L., ‘Periodic analogues of the Euler-Maclaurin and Poisson summation formulas with applications to number theory’, Acta Arith. 28 (1975), 23–68.CrossRefGoogle Scholar
[2]Berndt, B. C., ‘Periodic Bernoulli numbers, summation formulas and applications’, (Theory and application of specialfunctions, Academic Press, New York, 1975).Google Scholar
[3]Hardy, G. H., Divergent series (Oxford University Press, 1949).Google Scholar
[4]Hille, E. and Phillips, R. S., Functional analysis and semi-groups (Amer. Math. Soc. Colloquium Publ. 31, Providence, R. I., 1957).Google Scholar
[5]Miller, J. B., ‘The Euler-Maclaurin sum formula for an inner derivation’, Aequationes Math. 25 (1982), 42–51.CrossRefGoogle Scholar
[6]Miller, J. B., ‘The Euler-Maclaurin sum formula for a closed derivation’, J. Austral. Math. Soc. Series A 37 (1984), 128–138.CrossRefGoogle Scholar
[7]Miller, J. B., ‘The Euler-Maclaurin formula generated by a summation operator’, Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), 285–300.CrossRefGoogle Scholar
[8]Miller, J. B., ‘Series like Taylor's series’, Aequationes Math. 26 (1983), 208–220.CrossRefGoogle Scholar
[9]Miller, J. B., ‘The operator remainder in the Euler-Maclaurin formula’, A equationes Math. 28 (1985), 64–68.CrossRefGoogle Scholar
[10]Widder, D. V., The Laplace transform (Princeton Univ. Press, 1946).Google Scholar
[11]Walter, W., ‘Remark’, A equationes Math. 26 (1983), 281–282.Google Scholar