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Tempered Distributions Supported on a Half-Space of RN and Their Fourier Transforms

Published online by Cambridge University Press:  20 November 2018

Jean-Pierre Gabardo
Jean-Pierre Gabardo*
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1
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A fundamental problem in Fourier analysis is to characterize the behaviour of a function (or distribution) whose Fourier transform vanishes in some particular set. Of course, this is, in general, a very difficult question and little seems to be known, except in some special cases. For example, a theorem of Paley-Wiener (Theorem XII in [6]) characterizes exactly the behaviour of the modulus of a function in L2(R) whose Fourier transform vanishes on a half-line.

Keywords

42B1046F10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 1 , 01 February 1991 , pp. 61 - 88
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Beurling, A., On two problems concerning linear transformations in Hilbert space, Acta Math. 81(1949), 239–255.Google Scholar
2. Grenander, U. and Szegö, G., Toeplitz forms and their applications, University of California Press, 1958.Google Scholar
3. Helson, H., Lectures on invariant subspaces. Academic Press Inc.,New York, 1964.Google Scholar
4. Helson, H. and Lowdenslager, D., Prediction theory and Fourier series in several variables, Acta Math. 99(1958), 165–202.Google Scholar
5. Hoffman, K., Banach spaces of analytic functions. Prentice Hall, Englewood Cliffs, N. J., 1962.Google Scholar
6. Paley, R. and Wiener, N., Fourier transforms in the complex domain. A. M. S. Colloquium Publications, 19, New-York, 1934.Google Scholar
7. Rudin, W., Fourier analysis on groups. Interscience Publishers, Inc.,New York, 1962.Google Scholar
8. Rudin, W., Real and Complex analysis. 3d ed.,McGraw-Hill Book Company, New-York, 1987.Google Scholar
9. Schwartz, L., Théorie des distributions. Hermann, Paris, 1966.Google Scholar
10. Shapiro, H.S., Bounded functions with one-sided spectral gaps, Michigan Math J. 19(1972), 167–172.Google Scholar
11. Shapiro, H.S., Functions with a spectral gap, Bull. Amer. Math. Soc. 79(1973), 355–360.Google Scholar
12. Szegö, G., Beiträge zur théorie der toeplitzschen Formen (Erste Mitteilung), Math. Z. 6(1920), 167–202.Google Scholar