Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T02:21:53.475Z Has data issue: false hasContentIssue false
  • English
  • Français

A tauberian theorem related to the modified Hankel transform

Published online by Cambridge University Press:  17 April 2009

Kusum Soni  and
R.P. Soni
Kusum Soni
Affiliation:
Department of Mathematics, University of Tennessee, Knoxvilie, Tennessee, USA.
R.P. Soni
Affiliation:
Department of Mathematics, University of Tennessee, Knoxvilie, Tennessee, USA.
Rights & Permissions [Opens in a new window]

Abstract

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 modified Hankel transform arises naturally in connection with certain semigroup operations on measures in probability theory. We give a tauberian theorem for this transform when certain higher moments exist. The probabilistic significance of our result is that it translates a regularity condition on the transform into a direct condition on the measure. This complements earlier results by Pitman and Bingham for the trigonometric and the modified Hankel transform respectively.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 11 , Issue 2 , October 1974 , pp. 167 - 180
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Adamović, Dušan D., “Sur quelques propriétés des fonctions á croissance lente de Karamata. I”, Mat. Vesnik 3 (18) (1966), 123–136.Google Scholar
[2]Adamović, Dušan D., “Sur quelques propriétés des fonctions á croissance lente de Karamata. II”, Mat. Vesnik 3 (18) (1966), 161–172.Google Scholar
[3]Aljančić, S., Bojanič, R. et Tomić, M., “Sur la valeur asymptotique d'une classe des integrales definies”, Acad. Serbe Sci. Publ. Inst. Math. 7 (1954), 81–94.Google Scholar
[4]Bingham, N.H., “A Tauberian theorem for integral transforms of Hankel type”, J. London Math. Soc. (2) 5 (1972), 493–503.CrossRefGoogle Scholar
[5]Bojanic, R. and Karamata, J., “On slowly varying functions and asymptotic relations”, MRC Technical Report No. 432 (University of Wisconsin, Madison, 1963).Google Scholar
[6]Erdélyi, Arthur, Magnus, Wilhelm, Oberhettinger, Fritz, Tricomi, Francesco G. (edited by), Higher transcendental functions, Volume II. Based, in part, on notes left by Harry Bateman. (McGraw-Hill, New York, Toronto, London, 1953.)Google Scholar
[7]Erdélyi, Arthur, Magnus, Wilhelm, Oberhettinger, Fritz, Tricomi, Francesco G. (edited by), Tables of integral transforms, Volume I. Based, in part, on notes left by Harry Bateman. (McGraw-Hill, New York, Toronto, London, 1953.)Google Scholar
[8]Pitman, E.J.G., “On the behaviour of the characteristic function of a probability distribution in the neighbourhood of the origin”, J. Austral. Math. Soc. 8 (1968), 423–443.CrossRefGoogle Scholar
[9]Soni, K. and Soni, R.P., “Lipschitz behavior and characteristic functions”, SIAM J. Math. Anal. 4 (1973), 302–308.CrossRefGoogle Scholar
[10]Soni, K. and Soni, R.P., “Integrability theorems for a class of integral transforms“, J. Math. Anal. Appl. 43 (1973), 397–418.CrossRefGoogle Scholar
[11]Soni, K. and Soni, R.P., “A note on probability distribution functions”, submitted.Google Scholar
[12]Titchmarsh, E.C., The theory of functions, 2nd ed. (Clarendon Press, Oxford, 1939).Google Scholar
[13]Titchmarsh, E.C., Introduction to the theory of Fourier integrals 2nd ed. (Clarendon Press, Oxford, 1948).Google Scholar