The Case of Compact Symmetric Spaces

  • Valery V. VolchkovEmail author
  • Vitaly V. Volchkov
Part of the Springer Monographs in Mathematics book series (SMM)


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.


Symmetric Space Spherical Function Beltrami Operator Continuous Family Holomorphic Extension 
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Copyright information

© Springer-Verlag London Limited 2009

Authors and Affiliations

  1. 1.Mathematical DepartmentDonetsk National UniversityDonetskUkraine

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