Integration of positive linear functionals on a sphere in $R^{2n}$ with respect to Gaussian surface measures

Abstract

In this paper we present a formula for the calculation of the integrals of the form $\int_S u^{\ast}Xu\,\nu(\deri u)$, where $S$ is the unit sphere in $\mathbb{R}^{N}$, $X$ is a positive semi-definite symmetric matrix, and $\nu$ is a surface measure generated by a Gaussian measure $\mu$. The solution has the form $\mathrm{trace}(XZ)$, with the explicit procedure for the calculation of the matrix $Z$ which does not depend on $X$.