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Derivatives of isotropic positive definite functions on spheres


Authors: Mara Trübner and Johanna F. Ziegel
Journal: Proc. Amer. Math. Soc. 145 (2017), 3017-3031
MSC (2010): Primary 43A35, 33C50, 33C55, 60E10
DOI: https://doi.org/10.1090/proc/13561
Published electronically: January 25, 2017
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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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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

DOI: https://doi.org/10.1090/proc/13561
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
Article copyright: © Copyright 2017 American Mathematical Society