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Distributional Watson Transforms

Published online by Cambridge University Press:  20 November 2018

Hsing-Yuan Hsu
Hsing-Yuan Hsu*
Affiliation:
University of Washington, Seattle, Washington
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All our notation is as denned in [2] with the restriction to n = 1. However, for our purposes, we introduce a sequence of norms by

in It is not difficult to see that turns out to be a fundamental space.

It is a well-known fact that the Watson transform and the Mellin transform are connected by the fact that

and

if and only if K(s)K(l — s) = 1, where K(s) is the Mellin transform of k(x). Further, the Hankel transform and Hilbert transform can be considered as special cases of Watson transforms.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 6 , December 1972 , pp. 1191 - 1197
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Titchmarsch, E. C., Introduction to the theory of Fourier integrals. 2nd edition (Oxford University Press, 1948).Google Scholar
2. Zemanian, A., The distributional Laplace and Mellin transformations, SI AM J. Appl. Math. 14 (1966), 41–59.Google Scholar