Generalized Watson transforms I: General theory

Zheng, Qifu

doi:10.1090/S0002-9939-00-05399-5

Generalized Watson transforms I: General theory
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Proc. Amer. Math. Soc. 128 (2000), 2777-2787 Request permission

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

Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
G. Hardy and E. Titchmarsh, A class of Fourier kernels. Proc. London Math. Soc., (2), 35 (1933), 116–155.
R. A. Kunze, Some problems in analysis related to representation theory, preprint (1997), to appear
E. C. Titchmarsh, Introduction to the theory of Fourier integrals, 3rd ed., Chelsea Publishing Co., New York, 1986. MR 942661
G.N. Watson, General transforms, Proc. London Math. Soc., (2), 35 (1933), 156–199.
Q. Zheng, Generalized Watson transforms II: A new construction of complementary series of $GL(2, R)$ and properties of Bessel functions, in preparation.
Q. Zheng, Generalized Watson transforms III: Hankel transforms on symmetric cones, in preparation.