Convolution roots and differentiability of isotropic positive definite functions on spheres

Ziegel, Johanna

doi:10.1090/S0002-9939-2014-11989-7

Convolution roots and differentiability of isotropic positive definite functions on spheres
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Proc. Amer. Math. Soc. 142 (2014), 2063-2077 Request permission

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.

References

Sudipto Banerjee, On geodetic distance computations in spatial modeling, Biometrics 61 (2005), no. 2, 617–625. MR 2140936, DOI 10.1111/j.1541-0420.2005.00320.x
R. K. Beatson, W. zu Castell, and Y. Xu, A Pólya criterion for (strict) positive definiteness on the sphere, IMA J. Numer. Anal. (2013), DOI 10.1093/imanum/drt008.
R. P. Boas Jr. and M. Kac, Inequalities for Fourier transforms of positive functions, Duke Math. J. 12 (1945), 189–206. MR 12152
Roberto Cavoretto and Alessandra De Rossi, Fast and accurate interpolation of large scattered data sets on the sphere, J. Comput. Appl. Math. 234 (2010), no. 5, 1505–1521. MR 2610367, DOI 10.1016/j.cam.2010.02.031
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
Digital Library of Mathematical Functions, Release date 2012-03-23, National Institute of Standards and Technology from http://dlmf.nist.gov/, 2011.
Werner Ehm, Tilmann Gneiting, and Donald Richards, Convolution roots of radial positive definite functions with compact support, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4655–4685. MR 2067138, DOI 10.1090/S0002-9947-04-03502-0
Anne Estrade and Jacques Istas, Ball throwing on spheres, Bernoulli 16 (2010), no. 4, 953–970. MR 2759164, DOI 10.3150/09-BEJ241
Gregory E. Fasshauer and Larry L. Schumaker, Scattered data fitting on the sphere, Mathematical methods for curves and surfaces, II (Lillehammer, 1997) Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 1998, pp. 117–166. MR 1640548
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
L. V. Hansen, T. L. Thorarinsdottir, and T. Gneiting, Lévy particles: Modelling and simulating star-shaped random sets, CSGB Research Report (2011).
Chunfeng Huang, Haimeng Zhang, and Scott M. Robeson, On the validity of commonly used covariance and variogram functions on the sphere, Math. Geosci. 43 (2011), no. 6, 721–733. MR 2824128, DOI 10.1007/s11004-011-9344-7
G. Matheron, 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
Allan Pinkus, Strictly Hermitian positive definite functions, J. Anal. Math. 94 (2004), 293–318. MR 2124464, DOI 10.1007/BF02789051
Manfred Reimer, Multivariate polynomial approximation, International Series of Numerical Mathematics, vol. 144, Birkhäuser Verlag, Basel, 2003. MR 2003508, DOI 10.1007/978-3-0348-8095-4
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, DOI 10.1215/S0012-7094-42-00908-6
Michael Schreiner, Locally supported kernels for spherical spline interpolation, J. Approx. Theory 89 (1997), no. 2, 172–194. MR 1447837, DOI 10.1006/jath.1997.3037
S. Soubeyrand, J. Enjalbert, and I. Sache, Accounting for roughness of circular processes: Using Gaussian random processes to model the anisotropic spread of airborne plant disease, Theor. Popul. Biol. 73 (2008), 92–103.
A. Tovchigrechko and I. A. Vakser, How common is the funnel-like energy landscape in protein-protein interactions?, Protein Science 10 (2001), 1572–1583.
Dirk Werner, Funktionalanalysis, Third, revised and extended edition, Springer-Verlag, Berlin, 2000 (German, with German summary). MR 1787146
Andrew T. A. Wood, When is a truncated covariance function on the line a covariance function on the circle?, Statist. Probab. Lett. 24 (1995), no. 2, 157–164. MR 1354370, DOI 10.1016/0167-7152(94)00162-2
Yuan Xu and E. W. Cheney, Strictly positive definite functions on spheres, Proc. Amer. Math. Soc. 116 (1992), no. 4, 977–981. MR 1096214, DOI 10.1090/S0002-9939-1992-1096214-6
Johanna Ziegel, Stereological modelling of random particles, Comm. Statist. Theory Methods 42 (2013), no. 7, 1428–1442. MR 3031290, DOI 10.1080/03610926.2012.709299