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Convolution roots and differentiability of isotropic positive definite functions on spheres


Author: Johanna Ziegel
Journal: Proc. Amer. Math. Soc. 142 (2014), 2063-2077
MSC (2010): Primary 42A82, 33C50, 33C55, 60E10
Published electronically: February 28, 2014
MathSciNet review: 3182025
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Abstract: We prove that any isotropic positive definite function on the sphere can be written as the spherical self-convolution of an isotropic real-valued function. It is known that isotropic positive definite functions on $ d$-dimensional Euclidean space admit a continuous derivative of order $ [(d-1)/2]$. We show that the same holds true for isotropic positive definite functions on spheres and prove that this result is optimal for all odd dimensions.


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Additional Information

Johanna Ziegel
Affiliation: Department of Mathematics and Statistics, University of Bern, Institute of Mathematical Statistics and Actuarial Science, Sidlerstrasse 5, 3012 Bern, Switzerland
Email: johanna.ziegel@stat.unibe.ch

DOI: https://doi.org/10.1090/S0002-9939-2014-11989-7
Keywords: Convolution root, covariance function, isotropic positive definite function, radial basis function, spherical convolution, turning bands operator
Received by editor(s): July 4, 2012
Published electronically: February 28, 2014
Communicated by: Ken Ono
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.