Infinite dimensional Lapiacian and spherical harmonics

Abstract

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 $n\to \infty$, 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.