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The Case of Compact Symmetric Spaces

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

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag London Limited 2009

Authors and Affiliations

  1. 1.Mathematical DepartmentDonetsk National UniversityDonetskUkraine

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