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The Harish-Chandra Schwartz Space (HCS) and Anker’s Proof of Inversion

  • Jay Jorgenson
  • Serge Lang
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

As we have already mentioned, Harish-Chandra first set up his inversion theorem on a space which he defined, as an analogue of the Schwartz space in ordinary Fourier analysis. Classically, such analysis takes place on the imaginary axis, with the Fourier kernel e ixy 2. Under the Fourier transform, C c goes to the Paley-Wiener space. As one considers functions which are less restricted than C c , the image space changes accordingly, and the Fourier transforms are not necessarily entire. At a pivotal stage, the Schwartz space is self-dual, with functions on the given euclidean space or its corresponding imaginary axis iav. This leads naturally into the L2-duality. The Fourier transform of functions in spaces between the Schwartz space and C c although not entire may have analytic continuation to a domain larger than the imaginary axis, namely tube domains, which provide an added structure to the situation.

Keywords

Spherical Function Polynomial Growth Polar Decomposition Convolution Product Schwartz Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 2001

Authors and Affiliations

  • Jay Jorgenson
    • 1
  • Serge Lang
    • 2
  1. 1.Department of MathematicsCity College of New York, CUNYNew YorkUSA
  2. 2.Department of MathematicsYale UniversityNew HavenUSA

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