Publications of the Research Institute for Mathematical Sciences

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Volume 1, Issue 2, 1965, pp. 163–186
DOI: 10.2977/prims/1195196331

Published online: 1965-12-31

Infinite dimensional Lapiacian and spherical harmonics

Yasuo Umemura[1] and Norio Kono[2]

(1) Kyoto University, Japan
(2) Kyoto University, Japan

The purpose of the present paper is to discuss the properties of the Gaussian measure and the Lapiacian operator in infinite dimensions as limits of finite dimensional analogues. In the limit of n → ∞, the uniform measure on the n-dimensional sphere, the spherical Laplacian operator, and Gegenbauer polynomials (spherical harmonics) tend respectively to an infinite dimensional Gaussian measure, the infinite dimensional Laplacian operator, and Hermite polynomials. We also discuss the addition formula and integral representation formula of Hermite polynomials in this limit procedure.

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Umemura Yasuo, Kono Norio: Infinite dimensional Lapiacian and spherical harmonics. Publ. Res. Inst. Math. Sci. 1 (1965), 163-186. doi: 10.2977/prims/1195196331