A limit formula for semigroups defined by Fourier-Jacobi series

Abstract:

I. J. Schoenberg showed the following result in his celebrated paper [Schoenberg, I. J., Positive definite functions on spheres. Duke Math. J. 9 (1942), 96-108]: let $\cdot$ and $S^d$ denote the usual inner product and the unit sphere in $\mathbb {R}^{d+1}$, respectively. If $\mathcal {F}^d$ stands for the class of real continuous functions $f$ with domain $[-1,1]$ defining positive definite kernels $(x,y)\in S^d \times S^d \to f(x\cdot y)$, then the class $\bigcap _{d\geq 1} \mathcal {F}^d$ coincides with the class of probability generating functions on $[-1,1]$. In this paper, we present an extension of this result to classes of continuous functions defined by Fourier-Jacobi expansions with nonnegative coefficients. In particular, we establish a version of the above result in the case in which the spheres $S^d$ are replaced with compact two-point homogeneous spaces.