Generalized Watson transforms I: General theory

Author:
Qifu Zheng

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2777-2787

MSC (2000):
Primary 22E30, 43A32, 44A15; Secondary 43A65, 42A38

Published electronically:
February 29, 2000

MathSciNet review:
1670364

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Abstract | References | Similar Articles | Additional Information

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.

**1.**S. Bochner and K. Chandrasekharan,*Fourier Transforms*, Annals of Mathematics Studies, no. 19, Princeton University Press, Princeton, N. J.; Oxford University Press, London, 1949. MR**0031582****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. C. Titchmarsh,*Introduction to the theory of Fourier integrals*, 3rd ed., Chelsea Publishing Co., New York, 1986. MR**942661****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 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:
http://dx.doi.org/10.1090/S0002-9939-00-05399-5

Keywords:
Unitary representations,
intertwining operators,
Watson transform,
Hankel transform

Received by editor(s):
October 5, 1998

Published electronically:
February 29, 2000

Additional Notes:
This research was partially supported by National Science Foundation grant DMS-9501191

Communicated by:
Roe Goodman

Article copyright:
© Copyright 2000
American Mathematical Society