Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-13T18:22:27.568Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic behaviours of a class of integral transforms in complex domains

Published online by Cambridge University Press:  20 January 2009

R. S. Pathak  and
S. K. Mishra
R. S. Pathak
Affiliation:
Department of Mathematics, Banaras Hindu University, Varanasi 221 005, India
S. K. Mishra
Affiliation:
Department of Mathematics, Banaras Hindu University, Varanasi 221 005, India
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.

Zemanian [17] obtained abelian theorems for the Hankel and K-transforms of functions and then extended his results to the corresponding transforms of distributions in the sense of Schwartz [11]. Jones [6] has discussed at length asymptotic behaviours of transforms generalized in his sense. Following the technique of Zemanian many authors have obtained abelian theorems for more general transforms of functions and distributions in the sense of Schwartz. Mention may be made of the works of Joshi and Saxena [7], Lavoine and Misra [8] and Pathak [10]. However, these authors were confined to the transforms of real variables only.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 32 , Issue 2 , June 1989 , pp. 231 - 247
Copyright
Copyright © Edinburgh Mathematical Society 1989

References

REFERENCES

1.Bojanic, R. and Karamata, J., On slowly varying functions and asymptotic relations (MRC Technical Summary Report 432, October 1973, Madison, Wisconsin).Google Scholar
2.Carmichael, R. D. and Milton, E. O., Abelian theorems for the distributional Stieltjes transform, J. Math. Anal. Appl. 72 (1979), 195–205.Google Scholar
3.Carmichael, R. D. and Pathak, R. S., Asymptotic behaviour of the H-transform in the complex domains, Math. Proc. Cambridge Philos. Soc. 102 (1987), 533–552.CrossRefGoogle Scholar
4.Drozzinov, Ju. N. and Zav'jalov, B. I., The quasi-asymptotics of generalized functions and tauberian theorems in the complex domain, Math. USSR Sbornik 31 (1977) 329–345.Google Scholar
5.Gel'fand, I. M. and Shilov, G. E., Generalized functions, vol. I (Academic Press, New York, 1964).Google Scholar
6.Jones, D. S., Generalized transforms and their asymptotic behaviour, Philos. Trans. Roy. Soc. London Ser. A 265 (1969), 1–43.Google Scholar
7.Joshi, V. G. and Saxena, R. K., Abelian theorems for distributional H-transform, Math. Ann. 256 (1981), 311–321.CrossRefGoogle Scholar
8.Lavoine, 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
9.Luke, Y. L., The Special Functions and Their Approximations (Academic Press, New York, 1969).Google Scholar
10.Pathak, R. S., Abelian theorems for the G-transformation, J. Indian Math. Soc. 45 (1981), 243–249.Google Scholar
11.Schwartz, L., Théorie des Distributions (Hermann, Paris, 1966).Google Scholar
12.Stankovic, B., Abelian and tauberian theorems for Stieltjes transforms of distributions, Russian Math. Surveys 40:4 (1985), 99–133.Google Scholar
13.Takaci, A., A note on distributional Stieltjes transformation, Math. Proc. Cambridge Philos. Soc. 94 (1983), 523–527.Google Scholar
14.Treves, F., Topological vector spaces, Distributions and Kernels (Academic Press, New York, 1966).Google Scholar
15.Wimp, J., A class of integral transforms, Proc. Edinburgh Math. Soc. 14 (1964), 33–40.Google Scholar
16.Wong, R., Asymptotic expansions of the Kontorovich-Lebedev Transform, Applicable Anal. 12 (1981), 161–172.Google Scholar
17.Zemanian, A. H., Some abelian theorems for the distributional Hankel and K-transformations, SIAM J. Appl. Math. 14 (1966), 1255–1265.Google Scholar
18.Zemanian, A. H., Generalized Integral Transformations (Interscience Publishers, New York, 1968).Google Scholar