## 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.## References

- R. K. Beatson and W. zu Castell,
*Dimension hopping and families of strictly positive definite zonal basis functions on spheres*, preprint, arXiv:1510.08658 (2015). - Debao Chen, Valdir A. Menegatto, and Xingping Sun,
*A necessary and sufficient condition for strictly positive definite functions on spheres*, Proc. Amer. Math. Soc.**131**(2003), no. 9, 2733–2740. MR**1974330**, DOI 10.1090/S0002-9939-03-06730-3 - D. J. Daley and E. Porcu,
*Dimension walks and Schoenberg spectral measures*, Proc. Amer. Math. Soc.**142**(2014), no. 5, 1813–1824. MR**3168486**, DOI 10.1090/S0002-9939-2014-11894-6 - DLMF,
*NIST Digital Library of Mathematical Functions*, 2015, Release 1.0.10 of 2015-08-07. - G. Gaspari and S. E. Cohn,
*Construction of correlation functions in two and three dimensions*, Q. J. R. Meteorol. Soc.**125**(1999), 723–757. - Tilmann Gneiting,
*On the derivatives of radial positive definite functions*, J. Math. Anal. Appl.**236**(1999), no. 1, 86–93. MR**1702687**, DOI 10.1006/jmaa.1999.6434 - Tilmann Gneiting,
*Strictly and non-strictly positive definite functions on spheres*, Bernoulli**19**(2013), no. 4, 1327–1349. MR**3102554**, DOI 10.3150/12-BEJSP06 - Joseph Guinness and Montserrat Fuentes,
*Isotropic covariance functions on spheres: some properties and modeling considerations*, J. Multivariate Anal.**143**(2016), 143–152. MR**3431424**, DOI 10.1016/j.jmva.2015.08.018 - Annika Lang and Christoph Schwab,
*Isotropic Gaussian random fields on the sphere: regularity, fast simulation and stochastic partial differential equations*, Ann. Appl. Probab.**25**(2015), no. 6, 3047–3094. MR**3404631**, DOI 10.1214/14-AAP1067 - G. Matheron,
*Les variables régionalisées et leur estimation*, Masson et Cie, Paris, 1965. - —,
*Quelque aspects de la montée*, Note Géostatistique 120, Centre de Géostatistique, Fontainebleau, France, 1972. - Valdir A. Menegatto,
*Strictly positive definite kernels on the Hilbert sphere*, Appl. Anal.**55**(1994), no. 1-2, 91–101. MR**1379646**, DOI 10.1080/00036819408840292 - V. A. Menegatto, C. P. Oliveira, and A. P. Peron,
*Strictly positive definite kernels on subsets of the complex plane*, Comput. Math. Appl.**51**(2006), no. 8, 1233–1250. MR**2235825**, DOI 10.1016/j.camwa.2006.04.006 - I. J. Schoenberg,
*Metric spaces and completely monotone functions*, Ann. of Math. (2)**39**(1938), no. 4, 811–841. MR**1503439**, DOI 10.2307/1968466 - I. J. Schoenberg,
*Positive definite functions on spheres*, Duke Math. J.**9**(1942), 96–108. MR**5922** - Holger Wendland,
*Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree*, Adv. Comput. Math.**4**(1995), no. 4, 389–396. MR**1366510**, DOI 10.1007/BF02123482 - Zong Min Wu,
*Compactly supported positive definite radial functions*, Adv. Comput. Math.**4**(1995), no. 3, 283–292. MR**1357720**, DOI 10.1007/BF03177517 - A. M. Yaglom,
*Correlation theory of stationary and related random functions. Vol. I*, Springer Series in Statistics, Springer-Verlag, New York, 1987. Basic results. MR**893393** - Johanna Ziegel,
*Convolution roots and differentiability of isotropic positive definite functions on spheres*, Proc. Amer. Math. Soc.**142**(2014), no. 6, 2063–2077. MR**3182025**, DOI 10.1090/S0002-9939-2014-11989-7

## 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