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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.