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A tauberian theorem related to the modified Hankel transform

  • Kusum Soni (a1) and R.P. Soni (a1)

Abstract

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.

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References

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[1]Adamović, Dušan D., “Sur quelques propriétés des fonctions á croissance lente de Karamata. I”, Mat. Vesnik 3 (18) (1966), 123136.
[2]Adamović, Dušan D., “Sur quelques propriétés des fonctions á croissance lente de Karamata. II”, Mat. Vesnik 3 (18) (1966), 161172.
[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), 8194.
[4]Bingham, N.H., “A Tauberian theorem for integral transforms of Hankel type”, J. London Math. Soc. (2) 5 (1972), 493503.
[5]Bojanic, R. and Karamata, J., “On slowly varying functions and asymptotic relations”, MRC Technical Report No. 432 (University of Wisconsin, Madison, 1963).
[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.)
[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.)
[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), 423443.
[9]Soni, K. and Soni, R.P., “Lipschitz behavior and characteristic functions”, SIAM J. Math. Anal. 4 (1973), 302308.
[10]Soni, K. and Soni, R.P., “Integrability theorems for a class of integral transforms“, J. Math. Anal. Appl. 43 (1973), 397418.
[11]Soni, K. and Soni, R.P., “A note on probability distribution functions”, submitted.
[12]Titchmarsh, E.C., The theory of functions, 2nd ed. (Clarendon Press, Oxford, 1939).
[13]Titchmarsh, E.C., Introduction to the theory of Fourier integrals 2nd ed. (Clarendon Press, Oxford, 1948).
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A tauberian theorem related to the modified Hankel transform

  • Kusum Soni (a1) and R.P. Soni (a1)

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