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 be the Schwartz space of infinitely differentiable complex-valued functions defined on the real line with compact supports equipped with the usual topology. Assume to be the space of functions defined on the real line whose every element is the Hilbert transform of an element of . We equip the space with an appropriate topology and show that the classical Hilbert transformation , defined by , is a homeomorphism from onto . The Hilbert transform of is then defined to be an element of given by the relation

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

**[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)**

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