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The Hilbert transform of Schwartz distributions. II

Published online by Cambridge University Press:  24 October 2008

M. Aslam Chaudhry  and
J. N. Pandey
M. Aslam Chaudhry
Affiliation:
Department of Mathematics, University of Petroleum and Minerals, Dhahran, Saudi Arabia
J. N. Pandey
Affiliation:
Department of Mathematics and Statistics, Carleton University, Ottawa, Ontario KIS 5B6, Canada
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Abstract

Let D(R) be the Schwartz space of C∞ functions with compact support on R and let H(D) be the space of all C∞ functions defined on R for which every element is the Hilbert transform of an element in D(R), i.e.

where the integral is defined in the Cauchy principal-value sense. Introducing an appropriate topology in H(D), Pandey [3] defined the Hilbert transform Hf of f ∈ (D(R))′ as an element of (H(D))′ by the relation

and then with an appropriate interpretation he proved that

.

In this paper we give an intrinsic description of the space H(D) and its topology, thereby providing a solution to an open problem posed by Pandey ([4], p. 90).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 3 , November 1987 , pp. 553 - 559
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

[1]Ehrenpreis, L.. Analytic functions and the Fourier transform of distributions. I. Ann. of Math. 63 (1956), 120–159.CrossRefGoogle Scholar
[2]Friedman, A.. Generalized Functions and Partial Differential Equations (Prentice-Hall, 1963).Google Scholar
[3]Pandey, J. N. and Chaudhry, M. A.. The Hilbert transform of generalized functions and applications. Canad. J. Math. 35 (1983), 478–495.CrossRefGoogle Scholar
[4]Pandey, J. N.. The Hilbert transform of Schwartz distributions. Proc. Amer. Math. Soc. 89 (1983), 86–90.CrossRefGoogle Scholar
[5]Schwartz, L.. Théorie des distributions (Hermann, 1966).Google Scholar
[6]Titchmarsh, E. C.. Introduction to the Theory of Fourier Integrals (Oxford University Press, 1967).Google Scholar
[7]Zemanian, A. H.. Distribution Theory and Transform Analysis (McGraw-Hill, 1965).Google Scholar
[8]Zemanian, A. H., Generalized Integral Transforms (Interscience Publishers, 1968).Google Scholar