Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group pp 335-370 | Cite as

# The Case of Compact Symmetric Spaces

## Abstract

It is shown how the results of the previous chapter can be extended to compact symmetric spaces *U*/*K* of rank one. New features arise in this case. For example, the set of spherical functions is discrete. Accordingly, continuous analogues of the spherical transform and corresponding Paley–Wiener-type theorems are needed. This requires a detailed discussion of local eigenfunctions of the Laplace–Beltrami operator. Various properties of these eigenfunctions are presented at the beginning of the chapter. Then transmutation operators
\(\mathfrak{A}_{k,m,j}\)
are investigated. They are defined in a ball of *U*/*K* and are closely related to the Jacobi polynomial expansion. The last section is devoted to the study of analogues of
\(\mathfrak{A}_{k,m,j}\)
in the exterior of a ball.

## Keywords

Symmetric Space Spherical Function Beltrami Operator Continuous Family Holomorphic Extension## Preview

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