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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
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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.

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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

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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