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The Hilbert Transform of Generalized Functions and Applications

Published online by Cambridge University Press:  20 November 2018

J. N. Pandey  and
Muhammad Aslam Chaudhry
J. N. Pandey
Affiliation:
Carleton University, Ottawa, Ontario
Muhammad Aslam Chaudhry
Affiliation:
Carleton University, Ottawa, Ontario
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The theory of Fourier transforms of tempered distributions as developed by Laurent Schwartz [17] is quite simple and elegant and has wide variety of applications, but there does not exist a corresponding neat and simple theory for the Hilbert transform of generalized functions (distributions) having wide applications. One of the objectives of this paper is to develop such a theory for the Hilbert transform of generalized functions and indicate its applicability to a variety of problems.

In problems of physics sometimes we need to find harmonic functions u(x, y) in the region y > 0 whose limit as y → 0+ does not exist in pointwise sense but does exist in the distributional sense. The theory of Hilbert transform of generalized functions that we are going to develop will provide an answer to the existence and uniqueness of this problem.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 3 , 01 June 1983 , pp. 478 - 495
Copyright
Copyright © Canadian Mathematical Society 1983

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