The Helgason Fourier transform for compact Riemannian symmetric spaces of rank one
type: Item
instance of: scholarly article
title: The Helgason Fourier transform for compact Riemannian symmetric spaces of rank one
zbMATH Open document ID: 0707.43001
DOI: 10.1007/BF02392752
author: Thomas O. Sherman
published in: Acta Mathematica
publication date: 1990-01-01T00:00:00Z
review text: Since Helgason's famous papers it is well-known, that on the Riemannian symmetric spaces of the non-compact type there is a beautiful theory of a Fourier transform which in many aspects is analogous to the theory of the classical Fourier transform in the Euclidean space; for a detailed treatment see \textit{S. Helgason}'s book ``Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions.'' (1984; Zbl 0543.58001). In particular, the Helgason Fourier transform is defined by a kernel of a similar simplicity as the exponential function in the Euclidean case and there are simple inversion formulas and Plancherel theorems. Of course, the Helgason theory is closely related to the representation theory approach to harmonic analysis but it has the advantage of having nice relations to the Riemannian geometry of the underlying spaces. On the Riemannian symmetric spaces of the compact type harmonic analysis is very well-known through the representation theoretical approach. So it is surprising that no analog of the geometrical Helgason Fourier transform for the Riemannian symmetric spaces of the compact type was known. Now the aim of the present, very clearly written, paper is to close this gap at least for the compact rank 1 spaces. To develop such a theory the main difficulty is to find the kernel of the Fourier transform. In contrast to the noncompact case it involves a singularity caused by the existence of closed geodesics in the compact symmetric spaces. A local theory avoiding any problem with this singularity can be developed by restricting the support of the functions to be transformed to a sufficiently small open neighbourhood of the identity class in G/K. However, the global theory is much more difficult. Note that the local theory can be developed without restricting to the rank 1 case, but for the global theory the rank 1 condition is still necessary. The definition of the global Fourier transform kernel is based on…
Mathematics Subject Classification ID: 43A25
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 4161399
zbMATH Keywords: inversion formulas
zbMATH Keywords: Plancherel theorems
zbMATH Keywords: Riemannian symmetric spaces of the compact type
zbMATH Keywords: geometrical Helgason Fourier transform
zbMATH Keywords: global Fourier transform kernel
Wikidata QID: Q115392309
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02392752
OpenAlex ID: W2058568741
description: scientific article; zbMATH DE number 4161399