Hostname: page-component-848d4c4894-p2v8j Total loading time: 0 Render date: 2024-06-11T21:22:26.589Z Has data issue: false hasContentIssue false
  • English
  • Français

The Euler-Maclaurin sum formula for a closed derivation

Part of: Special classes of linear operators Approximations and expansions

Published online by Cambridge University Press:  09 April 2009

John Boris Miller
John Boris Miller
Affiliation:
Department of Mathematics Monash UniversityClayton, 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.

An operator form of the Euler-Maclaurin sum formula is obtained, expressing the sum of the Euler-Maclaurin infinite series in a closed derivation, whose spectrum is compact, not equal to {0}, and does not have 0 as a clusterpoint, as the difference between a summation operator and an antiderivation which is the local inverse of the derivation.

Keywords

Euler-Maclaurin formuladerivationantiderivationsummation operatorBaxter operator.

MSC classification

Secondary: 47B47: Commutators, derivations, elementary operators, etc. 41A58: Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 37 , Issue 1 , August 1984 , pp. 128 - 138
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Atkinson, F. V., ‘Some aspects of Baxter's functional equation’, J. Math. Anal. Appl. 7 (1963), 1–30.CrossRefGoogle Scholar
[2]Baxter, G., ‘An analytic problem whose solution follows from a simple algebraic identity’, Pacific J. Math. 10 (1960), 731–742.CrossRefGoogle Scholar
[3]Bong, N.-H., ‘Some apparent connection between Baxter and averaging operators’, J. Math. Anal. Appl. 56 (1976), 330–345.CrossRefGoogle Scholar
[4]Hille, E. and Phillips, R. S., Functional analysis and semi-groups (Amer. Math. Soc., Providence, R. I., 1957).Google Scholar
[5]Kingman, J. F. C., ‘Spitzer's identity and its use in probability theory,’ J. London Math. Soc. 37 (1962), 309–316.CrossRefGoogle Scholar
[6]Miller, J. B., ‘Some properties of Baxter operators’, Acta Math. Acad. Sci. Hungar. 17 (1966), 387–400.CrossRefGoogle Scholar
[7]Miller, J. B., ‘Baxter operators and endomorphisms on Banach algebras’, J. Math. Anal. Appl. 25 (1969), 503–520.CrossRefGoogle Scholar
[8]Miller, J. B., ‘The Euler-Maclaurin sum formula for an inner derivation’, Aequationes Math. 25 (1982), 42–51.CrossRefGoogle Scholar
[9]Rota, G.-C., ‘Baxter operators and combinatorial identities I and II’, Bull. Amer. Math. Soc. 75 (1969), 325–329 and 330–334.CrossRefGoogle Scholar
[10]Rota, G.-C. and Smith, D. A., ‘Fluctuation theory and Baxter algebras’, Symposia Mathematica, Vol. 9, 179–201 (Academic Press, 1972).Google Scholar
[11]Whittaker, E. T. and Watson, G. N., A course in modern analysis, Cambridge University Press, 1946.Google Scholar
[12]Miller, J. B., ‘The Euler-Maclaurin formula generated by a summation operator’, Proc. Royal Soc. Edinburgh 95A (1983), 285–300.CrossRefGoogle Scholar
[13]Miller, J. B., ‘The operator remainder in the Euler-Maclaurin formula’ (Analysis Paper 37, Monash University, Melbourne, 1983).Google Scholar
[14]Miller, J. B., ‘Series like Taylor's series’, Aequationes Math. (to appear).Google Scholar