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 S^d denote the usual inner product and the unit sphere in R d + 1 \mathbb {R}^{d+1} , respectively. If F d \mathcal {F}^d stands for the class of real continuous functions f f with domain [ − 1 , 1 ] [-1,1] defining positive definite kernels ( x , y ) ∈ S d × S d → f ( x ⋅ y ) (x,y)\in S^d \times S^d \to f(x\cdot y) , then the class ⋂ d ≥ 1 F d \bigcap _{d\geq 1} \mathcal {F}^d coincides with the class of probability generating functions on [ − 1 , 1 ] [-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 S^d are replaced with compact two-point homogeneous spaces.