Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-20T14:26:23.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic behaviour of the H-transform in the complex domain

Published online by Cambridge University Press:  24 October 2008

Richard D. Carmichael  and
Ram S. Pathak
Richard D. Carmichael
Affiliation:
Wake Forest University, Winston-Salem, NC 27109, U.S.A.
Ram S. Pathak
Affiliation:
Banaras Hindu University, Varanasi 221005, India
Get access Rights & Permissions [Opens in a new window]

Abstract

Abelian theorems for the H-transform of functions and generalized functions are obtained as the complex variable of the transform approaches zero or infinity in a wedge domain in the right half plane. Quasi-asymptotic behaviour (q.a.b.) of the H-transformable generalized functions is defined. A structure theorem for generalized functions possessing q.a.b. is proved and is applied to obtain the asymptotic behaviour of the H-transform of generalized functions having q.a.b. The theorems are illustrated by examples.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 3 , November 1987 , pp. 533 - 552
Copyright
Copyright © Cambridge Philosophical Society 1987

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]Boas, R. P.. Integrability Theorems for Trigonometric Transforms (Springer-Verlag, 1967).CrossRefGoogle Scholar
[2]Braaksma, B. L. J.. Asymptotic expansions and analytic continuations for a class of Barnes-integrals. Compositio Math. 15 (1964), 239–341.Google Scholar
[3]Carmichael, R. D. and Hayashi, E. K.. Abelian theorems for the Stieltjes transform of functions, II. Internat. J. Math. Math. Sci. 4 (1981), 67–88.CrossRefGoogle Scholar
[4]Carmichael, R. D. and Milton, E. O.. Abelian theorems for the distributional Stieltjes transform. J. Math. Anal. Appl. 72 (1979), 195–205.CrossRefGoogle Scholar
[5]Carmichael, R. D. and Pathak, R. S.. Abelian theorems for Whittaker transforms. Internat. J. Math. Math. Sci. 10 (1987), 417–431.CrossRefGoogle Scholar
[6]Doetsch, G.. Introduction to the Theory and Application of the Laplace Transformation (Springer-Verlag, 1974).CrossRefGoogle Scholar
[7]Drožžinov, Ju. N. and Zav'jalov, B. I.. Quasi-asymptotics of generalized functions and Tauberian theorems in the complex domain. Math. USSR-Sb. 31 (1977), 329–345.CrossRefGoogle Scholar
[8]Gel'fand, I. M. and Shilov, G. E.. Generalized Functions, Volume I (Academic Press, 1964).Google Scholar
[9]Jones, D. S.. Generalized transforms and their asymptotic behaviour. Philos. Trans. Roy. Soc. London Ser. A 265 (1969), 1–43.Google Scholar
[10]Joshi, V. G. and Saxena, R. K.. Abelian theorems for distributional H-transform. Math. Ann. 256 (1981), 311–321.CrossRefGoogle Scholar
[11]Lavoine, J. and Misra, O. P.. Théorèmes abéliens pour la transformation de Stieltjes des distributions. C. R. Acad. Sci. Paris Ser. I. Math. 279 (1974), 99–102.Google Scholar
[12]Lavoine, J. and Misra, O. P.. Abelian theorems for the distributional Stieltjes transformation. Math. Proc. Cambridge Philos. Soc. 86 (1979), 287–293.CrossRefGoogle Scholar
[13]Malgonde, S. P. and Saxena, R. K.. An inversion formula for the distributional H-transformation. Math. Ann. 258 (1982), 409–417.CrossRefGoogle Scholar
[14]Mathai, A. M. and Saxena, R. K.. The H-function with Applications in Statistics and Other Disciplines (John Wiley and Sons, 1978).Google Scholar
[15]Misra, O. P.. Some Abelian theorems for the distributional Meijer-Laplace transformation. Indian J. Pure Appl. Math. 3 (1972), 241–247.Google Scholar
[16]Pathak, R. S.. Abelian theorems for the G-transformation. J. Indian Math. Soc. 45 (1981), 243–249.Google Scholar
[17]Saxena, R. K. (Jodhpur). Abelian theorems for distributional H-transform. Acta Mexicana Cienc. Tecn. 7 (1973), 66–76.Google Scholar
[18]Schwartz, L.. Théorie des distributions (Hermann, 1966).Google Scholar
[19]Sneddon, I. N.. The Use of Integral Transforms (McGraw-Hill Book Co., 1972).Google Scholar
[20]Srivastava, H. M., Gupta, K. C. and Goyal, S. P.. The H-functions of One and Two Variables with Applications (South Asian Publishers, 1982).Google Scholar
[21]Takači, A.. A note on the distributional Stieltjes transformation. Math. Proc. Cambridge Philos. Soc. 94 (1983), 523–527.CrossRefGoogle Scholar
[22]Treves, F.. Topological Vector Spaces, Distributions and Kernels (Academic Press, 1966).Google Scholar
[23]Widder, D. V.. The Laplace Transform (Princeton University Press, 1946).Google Scholar
[24]Zayed, A.. Asymptotic expansions of some integral transforms by using generalized functions. Trans. Amer. Math. Soc. 272 (1982), 785–802.CrossRefGoogle Scholar
[25]Zemanian, A. H.. Some Abelian theorems for the distributional Hankel and K transformations. SIAM J. Appl. Math. 14 (1966), 1255–1265.CrossRefGoogle Scholar
[26]Zemanian, A. H.. Generalized Integral Transformations (Interscience Publishers, 1968).Google Scholar