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A limit formula for semigroups defined by Fourier-Jacobi series


Authors: J. C. Guella and V. A. Menegatto
Journal: Proc. Amer. Math. Soc. 146 (2018), 2027-2038
MSC (2010): Primary 33C45, 42A16, 42A82, 42C10
DOI: https://doi.org/10.1090/proc/13889
Published electronically: December 4, 2017
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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
Email: jeanguella@gmail.com

V. A. Menegatto
Affiliation: Departamento de Matemática, ICMC-USP - São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil
Email: menegatt@icmc.usp.br

DOI: https://doi.org/10.1090/proc/13889
Keywords: Jacobi polynomials, limit formulas, positive definiteness, two-point homogeneous spaces, Fourier-Jacobi expansions
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
Article copyright: © Copyright 2017 American Mathematical Society

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