The Helgason Fourier transform for compact Riemannian symmetric spaces of rank one
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 a regularization process and the most difficult part of the proof of the main theorem - the (global) Fourier inversion formula - is devoted to make this handable. In fact, it turns out, that for this purpose the nice theory of a map \(\xi\) : G/K\(\to \Omega\) from the symmetric space to the closed unit ball in Euclidean space of dimension 3, 5 or 9 has to be developed. Moreover, the theory of the map \(\xi\) is of independent interest.
Riemannian symmetric spaces of the compact type
scientific article; zbMATH DE number 4161399
scientific article