Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The Hilbert transform of Schwartz distributions


Author: J. N. Pandey
Journal: Proc. Amer. Math. Soc. 89 (1983), 86-90
MSC: Primary 46F12; Secondary 44A15
DOI: https://doi.org/10.1090/S0002-9939-1983-0706516-2
MathSciNet review: 706516
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathcal{D}$ be the Schwartz space of infinitely differentiable complex-valued functions defined on the real line with compact supports equipped with the usual topology. Assume $ H(\mathcal{D})$ to be the space of $ {C^\infty }$ functions defined on the real line whose every element is the Hilbert transform of an element of $ \mathcal{D}$. We equip the space $ H(\mathcal{D})$ with an appropriate topology and show that the classical Hilbert transformation $ H$, defined by $ Hf = P\int_{ - \infty }^\infty {f(t)/(t - x)dt} $, is a homeomorphism from $ \mathcal{D}$ onto $ H(\mathcal{D})$. The Hilbert transform $ Hf$ of $ f \in \mathcal{D}'$ is then defined to be an element of $ H'(\mathcal{D})$ given by the relation

$\displaystyle \left\langle {Hf,\varphi } \right\rangle = \left\langle {f, - H\varphi } \right\rangle \forall \varphi \in H(\mathcal{D}).$

It then follows that - $ - {H^2}f/{\pi ^2} = f\forall f \in \mathcal{D}'$.

Applications of our results in solving some singular integral equations are also discussed.


References [Enhancements On Off] (What's this?)

  • [1] E. J. Beltrami and M. R. Wohlers, Distributional boundary value theorems and Hilbert transforms, Arch. Rational Mech. Anal. 18 (1965), 304-309. MR 0179611 (31:3858)
  • [2] Avner Friedman, Generalized functions and partial differential equations, Prentice-Hall, Englewood Cliffs, N.J., 1963. MR 0165388 (29:2672)
  • [3] I. M. Gel'fand and G. E. Shilov, Generalized functions, Vol. II, Academic Press, New York, 1968. MR 0230128 (37:5693)
  • [4] Dragisa Mitrovic, A Hilbert boundary-value problem, Math. Balkanica 1 (1971), 177-180. MR 0288576 (44:5773)
  • [5] Marion Orton, Hilbert transforms, Plemelj relations and Fourier transforms of distributions, SIAM J. Math. Anal. 4 (1973), 656-667. MR 0331051 (48:9386)
  • [6] J. N. Pandey and A. Chaudhary, Hilbert transform of generalized functions and applications, Canad. J. Math. (to appear) MR 2222329 (2006k:42014)
  • [7] L. Schwartz, Théorie des distributions, Hermann, Paris, 1966. MR 0209834 (35:730)
  • [8] E. C. Titchmarsh, Introduction to the theory of Fourier integrals, Oxford Univ. Press, 1967.
  • [9] F. Tricomi, Integral equations, Interscience, New York, 1957. MR 0094665 (20:1177)
  • [10] A. H. Zemanian, Distribution theory and transform analysis, McGraw-Hill, New York, 1965. MR 0177293 (31:1556)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46F12, 44A15

Retrieve articles in all journals with MSC: 46F12, 44A15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0706516-2
Keywords: Generalized integral transform, integral transform of distributions
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society