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Eigenspaces of the Laplace-Beltrami operator on a hyperboloid

Published online by Cambridge University Press:  22 January 2016

Jiro Sekiguchi
Jiro Sekiguchi*
Affiliation:
Department of Mathematics, Tokyo Metropolitan University
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Ever since S. Helgason [4] showed that any eigenfunction of the Laplace-Beltrami operator on the unit disk is represented by the Poisson integral of a hyperfunction on the unit circle, much interest has been arisen to the study of the Poisson integral representation of joint eigenfunctions of all invariant differential operators on a symmetric space X. In particular, his original idea of expanding eigenfunctions into K-finite functions has proved to be generalizable up to the case where X is a Riemannian symmetric space of rank one (cf. [4], [5], [11]). Presently, extension to arbitrary rank has been completed by quite a different formalism which views the present problem as a boundary-value problem for the differential equations. It should be recalled that along this line of approach a general theory of the systems of differential equations with regular singularities was successfully established by Kashiwara-Oshima (cf. [6], [7]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 79 , October 1980 , pp. 151 - 185
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1980

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