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Generalized Mellin Convolutions and their Asymptotic Expansions

Published online by Cambridge University Press:  20 November 2018

R. Wong  and
J. P. Mcclure
R. Wong
Affiliation:
University of Manitoba, Winnipeg, Manitoba
J. P. Mcclure
Affiliation:
University of Manitoba, Winnipeg, Manitoba
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A large number of important integral transforms, such as Laplace, Fourier sine and cosine, Hankel, Stieltjes, and Riemann- Liouville fractional integral transforms, can be put in the form

1.1

where f(t) and the kernel, h(t), are locally integrable functions on (0,∞), and x is a positive parameter. Recently, two important techniques have been developed to give asymptotic expansions of I(x) as x → + ∞ or x → 0+. One method relies heavily on the theory of Mellin transforms [8] and the other is based on the use of distributions [24]. Here, of course, the integral I(x) is assumed to exist in some ordinary sense.

If the above integral does not exist in any ordinary sense, then it may be regarded as an integral transform of a distribution (generalized function). There are mainly two approaches to extend the classical integral transforms to distributions.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 5 , 01 October 1984 , pp. 924 - 960
Copyright
Copyright © Canadian Mathematical Society 1984

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