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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A limit formula for semigroups defined by Fourier-Jacobi series
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by J. C. Guella and V. A. Menegatto PDF
Proc. Amer. Math. Soc. 146 (2018), 2027-2038 Request permission

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