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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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Derivatives of isotropic positive definite functions on spheres
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by Mara Trübner and Johanna F. Ziegel PDF
Proc. Amer. Math. Soc. 145 (2017), 3017-3031 Request permission

Abstract:

We show that isotropic positive definite functions on the $d$-dimensional sphere which are $2k$ times differentiable at zero have $2k+[(d-1)/2]$ continuous derivatives on $(0,\pi )$. This result is analogous to the result for radial positive definite functions on Euclidean spaces. We prove optimality of the result for all odd dimensions. The proof relies on montée, descente and turning bands operators on spheres which parallel the corresponding operators originating in the work of Matheron for radial positive definite functions on Euclidean spaces.
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Additional Information
  • Mara Trübner
  • Affiliation: Department of Mathematics and Statistics, Institute of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
  • Email: mara.truebner@hotmail.com
  • Johanna F. Ziegel
  • Affiliation: Department of Mathematics and Statistics, Institute of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
  • Email: johanna.ziegel@stat.unibe.ch
  • Received by editor(s): March 22, 2016
  • Received by editor(s) in revised form: March 23, 2016, and August 19, 2016
  • Published electronically: January 25, 2017
  • Communicated by: Mark M. Neerschaert
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 3017-3031
  • MSC (2010): Primary 43A35, 33C50, 33C55, 60E10
  • DOI: https://doi.org/10.1090/proc/13561
  • MathSciNet review: 3637950