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Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Generalized Watson transforms I: General theory

Author(s): Qifu Zheng
Journal: Proc. Amer. Math. Soc. 128 (2000), 2777-2787.
MSC (2000): Primary 22E30, 43A32, 44A15; Secondary 43A65, 42A38
Posted: February 29, 2000
MathSciNet review: 1670364
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Abstract | References | Similar articles | Additional information

Abstract:

This paper introduces two main concepts, called a generalized Watson transform and a generalized skew-Watson transform, which extend the notion of a Watson transform from its classical setting in one variable to higher dimensional and noncommutative situations. Several construction theorems are proved which provide necessary and sufficient conditions for an operator on a Hilbert space to be a generalized Watson transform or a generalized skew-Watson transform. Later papers in this series will treat applications of the theory to infinite-dimensional representation theory and integral operators on higher dimensional spaces.

References:

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S. Bochner and K. Chandrasekharan, Fourier Transforms, Princeton, 1949. MR 11:173d

2.
G. Hardy and E. Titchmarsh, A class of Fourier kernels. Proc. London Math. Soc., (2), 35 (1933), 116-155.

3.
R. A. Kunze, Some problems in analysis related to representation theory, preprint (1997), to appear

4.
E. Titchmarsh, Theory of Fourier Integrals, 3rd edition, Oxford, 1986. MR 89c:42002

5.
G.N. Watson, General transforms, Proc. London Math. Soc., (2), 35 (1933), 156-199.

6.
Q. Zheng, Generalized Watson transforms II: A new construction of complementary series of $GL(2, R)$ and properties of Bessel functions, in preparation.

7.
Q. Zheng, Generalized Watson transforms III: Hankel transforms on symmetric cones, in preparation.

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Additional Information:

Qifu Zheng
Affiliation: Department of Mathematics and Statistics, The College of New Jersey, P.O. Box 7718, Ewing, New Jersey 08628-0718
Email: zheng@tcnj.edu

DOI: 10.1090/S0002-9939-00-05399-5
PII: S 0002-9939(00)05399-5
Keywords: Unitary representations, intertwining operators, Watson transform, Hankel transform
Received by editor(s): October 5, 1998
Posted: February 29, 2000
Additional Notes: This research was partially supported by National Science Foundation grant DMS-9501191
Communicated by: Roe Goodman
Copyright of article: Copyright 2000, American Mathematical Society




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