Fourier transforms of spherical distributions on compact symmetric spaces

• Autores: Gestur Olafsson, Henrik Schlichtkrull
• Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 109, Nº 1, 2011, págs. 93-113
• Idioma: inglés
• Enlaces
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• Resumen

  • In our previous articles [27] and [28] we studied Fourier series on a symmetric space M=U/K of the compact type. In particular, we proved a Paley-Wiener type theorem for the smooth functions on M, which have sufficiently small support and are K-invariant, respectively K-finite. In this article we extend these results to K-invariant distributions on M. We show that the Fourier transform of a distribution, which is supported in a sufficiently small ball around the base point, extends to a holomorphic function of exponential type. We describe the image of the Fourier transform in the space of holomorphic functions. Finally, we characterize the singular support of a distribution in terms of its Fourier transform, and we use the Paley-Wiener theorem to characterize the distributions of small support, which are in the range of a given invariant differential operator. The extension from symmetric spaces of compact type to all compact symmetric spaces is sketched in an appendix.