Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-05T06:47:25.195Z Has data issue: false hasContentIssue false
  • English
  • Français

Abelian theorems for the distributional Stieltjes transformation

Published online by Cambridge University Press:  24 October 2008

Jean Lavoine  and
O. P. Misra
Jean Lavoine
Affiliation:
C.N.R.S., 17 Rue Vérollot F 94800 Villejuif, France
O. P. Misra
Affiliation:
Indian Institute of Technology, New Delhi
Get access Rights & Permissions [Opens in a new window]

Extract

0. Introduction. Asymptotic behaviour of distributions (Silva(8) and Benedetto (l)) plays a fundamental role in the analysis of singularities of generalized integral transforms. Such analysis in terms of Abelian theorems for the distributional Stieltjes transformation has been obtained by Misra (6) and Lavoine and Misra (3). To obtain his results Misra used some Abelian theorems (see Misra (6), theorems 3·1·1 and 4·1·1) for the Stieltjes transformation of functions which were essentially generalizations of the results given by Widder (9), pp. 183–185. The object of the present paper is to complete the work given by the authors in (3); Sections 2 and 3 of this paper are devoted to obtaining Abelian theorems concerning the distributional Stieltjes transformation in terms of the behaviour of this transform near the origin and at infinity.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 86 , Issue 2 , September 1979 , pp. 287 - 294
Copyright
Copyright © Cambridge Philosophical Society 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Benedetto, J. J.Analytic representation of generalized functions. Math. Z. 97 (1967), 303–319.CrossRefGoogle Scholar
(2)Gelfand, I. M. and Shilov, G. E.Generalized functions, vol. 1, chap. 1, sec. 3 (New York, Academic Press, 1964).Google Scholar
(3)Lavoine, J. and Misra, O. P.Théorèmes abéliens pour la transformation de Stieltjes des distributions. C. R. Acad. Sci. Paris 279 (1974), 99–102.Google Scholar
(4)Lavoine, J. and Misra, O. P. Distribution theory in transform analysis (to appear).Google Scholar
(5)Lojasiewicz, S.Sur la valeur d'une distribution en un point. Studia Math. 16 (1957), 1–36.CrossRefGoogle Scholar
(6)Misra, O. P.Some abelian theorems for distributional Stieltjes transformation. J. Math. Anal. Appl. 39 (1972), 590–599.CrossRefGoogle Scholar
(7)Schwartz, L.Théorie des distributions (Paris, Hermann, 1966).Google Scholar
(8)Sebastiao e Silva, J.Integrals and order of growth of distributions (Instituto Gulbenkian de Ciencia Lisboa, 35 (1964), 7124).Google Scholar
(9)Widder, D. V.The Laplace transform (Princeton University Press, N.J. 1946).Google Scholar