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Representation of Functions as Weierstrass- Transforms

Published online by Cambridge University Press:  20 November 2018

H.P. Heinig
H.P. Heinig*
Affiliation:
McMaster University, Hamilton, Ontario University of Alberta, Edmonton, Alberta
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The Weierstrass - respectively Weierstrass - Stieltjes transform of a function F(t) or μ(t) is defined by

1.1

and

1.2

for all x for which these integrals converge. In what follows we shall always assume that F(t) is Lebesgue integrable in every finite interval and that μ(t) is a function of bounded variation.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 10 , Issue 5 , December 1967 , pp. 711 - 722
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Cooper, J. L. B., Umkehrformeln fur Fourier Transformationen, Approximations - und Interpolationstheory Sonderdruck, I.S N.M. vol. 5 (1966) Birkhauser Verlag, Stuttgart.Google Scholar
2. B, J. L.. Cooper, The Representation of Functions as Laplace transforms. Math. Annalen 159 223-233 (1966).Google Scholar
3. Nessel, R. J., Uber die DarstellungholomorpherFunktionen durch Weierstras - und Weier strass - Stieltjes Intègrale; J. F. reine u. angew. Math. 218, (1966) (31- 50).Google Scholar
4. Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals; Oxford (1933).Google Scholar
5. Widder, D. V., The Laplace Transform; Princeton (1944).Google Scholar