The Harish-Chandra Schwartz Space (HCS) and Anker’s Proof of Inversion
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.
KeywordsSpherical Function Polynomial Growth Polar Decomposition Convolution Product Schwartz Space
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