## A limit formula for semigroups defined by Fourier-Jacobi series

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- by J. C. Guella and V. A. Menegatto PDF
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## 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.

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## Additional Information

**J. C. Guella**- Affiliation: Departamento de Matemática, ICMC-USP - São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil
- MR Author ID: 1141900
- Email: jeanguella@gmail.com
**V. A. Menegatto**- Affiliation: Departamento de Matemática, ICMC-USP - São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil
- MR Author ID: 358330
- ORCID: 0000-0002-4213-8759
- Email: menegatt@icmc.usp.br
- Received by editor(s): February 6, 2017
- Received by editor(s) in revised form: June 29, 2017
- Published electronically: December 4, 2017
- Additional Notes: The second author was supported in part by FAPESP, Grant 2016/09906-0
- Communicated by: Yuan Xu
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**146**(2018), 2027-2038 - MSC (2010): Primary 33C45, 42A16, 42A82, 42C10
- DOI: https://doi.org/10.1090/proc/13889
- MathSciNet review: 3767354