Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T19:39:08.779Z Has data issue: false hasContentIssue false
  • English
  • Français

The Hilbert Problem—A Distributional Approach

Published online by Cambridge University Press:  20 November 2018

M. A. Chaudhry  and
J. N. Pandey
M. A. Chaudhry
Affiliation:
Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
J. N. Pandey
Affiliation:
Department of Mathematics and Statistics, Carleton University, Ottawa, Ontario
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A distributional solution to the Hilbert problem in dimension > 1 is given.

Keywords

46F1244A15Hilbert transformHilbert problemdistributional Hilbert transform in n dimension
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 34 , Issue 3 , 01 September 1991 , pp. 321 - 328
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Barros-Neto, J., An introduction to the theory of distributions. Marcel Dekkar, Inc., New York, 1973.Google Scholar
2. Beltrami, E. J. and Wohlers, M. R., Distributional boundary values theorems and Hilbert transforms, Arch. Rational Mech. Anal. 18 (1965), 304–309.Google Scholar
3. Fefferman, C., Estimates for double Hilbert transforms , Studia Mathematica VXLVI(1972), 1–15.Google Scholar
4. Kokilashville, V. M., Singular integral operators in weighted spaces , Colloquia Mathematica Societatis Janos Bolyai, 35 Functions series operators Budapest-(Hungary) (1980), 707–714.Google Scholar
5. Lauwerier, H. A., The Hilbert problem for generalized functions, Arch. Rational Mech. Anal. 13 (1963), 157–166.Google Scholar
6. Orton, M., Hilbert transforms, Plemelj relations and Fourier transforms of distributions, SI AM J. Math. Anal. 4 (1973), 656–667.Google Scholar
7. Pandey, J. N. and Chaudhry, M. A., Hilbert transform of generalized functions and applications , Canad. J. Math. (3)XXXV, 478–495.Google Scholar
8 Pandey, J. N. and Chaudhry, M. A., The Hilbert transforms of Schwartz distributions II. Proc. of Cambridge Philosophical Society, England (Part 2)102(1987).Google Scholar
9. Pandey, J. N. and Singh, O. P., The characterisation of functions with Fourier transforms supported on orthants, submitted.Google Scholar
10. Schwartz, L., Théories des Distributions. Vol. I, II. Hermann, Paris, 1957, 1959.Google Scholar
11. Singh, O. P. and Pandey, J. N., The n-dimensional Hilbert transform of distributions—its inversion and applications , Canad. J. Math. (2)XLII(1990), 239–258.Google Scholar
12. Titchmarsh, E. C., Introduction to the theory of Fourier-integrals. Clarendon Press, Oxford, 1936; 2nd éd., 1948.Google Scholar
13. Zemanian, A. H., Distribution theory and transform analysis. McGraw-Hill, New York, 1965.Google Scholar