Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-07-06T08:48:32.676Z Has data issue: false hasContentIssue false
  • English
  • Français

An inversion theorem for the Stieltjes transform of distributions

Published online by Cambridge University Press:  20 January 2009

Stevan Pilipović
Stevan Pilipović
Affiliation:
Institute of Mathematics, 21000 Novi Sad, Yugoslavia
Rights & Permissions [Opens in a new window]

Extract

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.

The Stieltjes transformation of generalized functions was investigated by Benedetto[1], Zemanian [16], Misra [10] Pandey [11], Lavoine and Misra [7], and Erdélyi [5]. The inversion theorems for the Stieltjes transform of generalized functions, in their own approaches, were given in Benedetto [1], Zemanian [16], Pandey [11], Pathak [12] (he used Pande's approach) and Erdélyi [5].

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 29 , Issue 2 , June 1986 , pp. 171 - 185
Copyright
Copyright © Edinburgh Mathematical Society 1986

References

1Benedetto, J. J., Analytic representation of generalized functions, Math. Z. 97 (1967), 303–319.CrossRefGoogle Scholar
2Byrne, A. and Love, E. R., Complex inversion theorems for generalized Stieltjes transforms, J. Australian Math. Soc. 18 (1974), 328–358.CrossRefGoogle Scholar
3Carmichael, R. D. and Milton, E. O., Abelian Theorems for the distributional Stieltjes transforms, J. Math. Anal. Appl. 11 (1979), 195–205.CrossRefGoogle Scholar
4Geltand, I. M. and Shilov, G. E., Generalized Functions, Vol. 1 (Academic Press, New York, 1964).Google Scholar
5Erdelyi, A., Stieltjes transform of generalized functions, Proc. Royal Soc. Edinburgh 76A (1977), 231–249.CrossRefGoogle Scholar
6Hirschman, I. I. and Widder, D. V., The convolution transform (Princeton University Press, Princeton, 1955).Google Scholar
7Lavoine, J. and Misra, O. P., Théorèmes abéliens pour la transformation de Stieltjes des distributions, C.R. Acad. Sci. Paris, Serie A 279 (1974), 99–102.Google Scholar
8Lavoine, J. and Misra, O. P., Abelian theorems for the distributional Stieltjes Transformation, Math. Proc. Camb. Philos. Soc. 86 (1979), 287–293.CrossRefGoogle Scholar
9Love, E. R. and Byrne, A., Real inversion theorems for generalized Stieltjes transforms, J. London Math. Soc. (2) 22 (1980), 285–306.CrossRefGoogle Scholar
10Misra, O. P., Some Abelian theorems for distributional Stieltjes transformation, J. Math. Anal. Appl. 39 (1972), 590–599.CrossRefGoogle Scholar
11Pandey, J. N., On the Stieltjes transform of generalized functions, Math. Proc. Camb. Philos. Soc. 71 (1971), 85–96.CrossRefGoogle Scholar
12Pathak, R. S., A distributional generalized Stieltjes transformation, Proc. Edinburgh Math. Soc. 20 (1976), 15–22.CrossRefGoogle Scholar
13Sumner, D. B., An inversion formula for the generalized Stieltjes transform, Bui. Amer. Math. Soc. 55 (1949), 174–183.CrossRefGoogle Scholar
14Vladimirov, V. S., Generalized Functions in Mathematical Physics (MIR Publishers, Moscow, 1979).Google Scholar
15Widder, D. V., The Laplace Transform (Princeton University Press, Princeton, N. J., 1946).Google Scholar
16Zemanian, A. H., Generalized Integral Transformations (Interscience, New York, 1968).Google Scholar