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

  • John Boris Miller (a1)

Abstract

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.

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References

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[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), 2368.
[2]Berndt, B. C., ‘Periodic Bernoulli numbers, summation formulas and applications’, (Theory and application of specialfunctions, Academic Press, New York, 1975).
[3]Hardy, G. H., Divergent series (Oxford University Press, 1949).
[4]Hille, E. and Phillips, R. S., Functional analysis and semi-groups (Amer. Math. Soc. Colloquium Publ. 31, Providence, R. I., 1957).
[5]Miller, J. B., ‘The Euler-Maclaurin sum formula for an inner derivation’, Aequationes Math. 25 (1982), 4251.
[6]Miller, J. B., ‘The Euler-Maclaurin sum formula for a closed derivation’, J. Austral. Math. Soc. Series A 37 (1984), 128138.
[7]Miller, J. B., ‘The Euler-Maclaurin formula generated by a summation operator’, Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), 285300.
[8]Miller, J. B., ‘Series like Taylor's series’, Aequationes Math. 26 (1983), 208220.
[9]Miller, J. B., ‘The operator remainder in the Euler-Maclaurin formula’, A equationes Math. 28 (1985), 6468.
[10]Widder, D. V., The Laplace transform (Princeton Univ. Press, 1946).
[11]Walter, W., ‘Remark’, A equationes Math. 26 (1983), 281282.
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The standard summation operator, the Euler-Maclaurin sum formula, and the Laplace transformation

  • John Boris Miller (a1)

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