### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 043, 19 pages      arXiv:1601.07743      https://doi.org/10.3842/SIGMA.2016.043
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

### One-Step Recurrences for Stationary Random Fields on the Sphere

R.K. Beatson a and W. zu Castell bc
a) School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand
b) Scientific Computing Research Unit, Helmholtz Zentrum München, Ingolstädter Landstraße 1, 85764 Neuherberg, Germany
c) Department of Mathematics, Technische Universität München, Germany

Received January 28, 2016, in final form April 15, 2016; Published online April 28, 2016

Abstract
Recurrences for positive definite functions in terms of the space dimension have been used in several fields of applications. Such recurrences typically relate to properties of the system of special functions characterizing the geometry of the underlying space. In the case of the sphere ${\mathbb S}^{d-1} \subset {\mathbb R}^d$ the (strict) positive definiteness of the zonal function $f(\cos \theta)$ is determined by the signs of the coefficients in the expansion of $f$ in terms of the Gegenbauer polynomials $\{C^\lambda_n\}$, with $\lambda=(d-2)/2$. Recent results show that classical differentiation and integration applied to $f$ have positive definiteness preserving properties in this context. However, in these results the space dimension changes in steps of two. This paper develops operators for zonal functions on the sphere which preserve (strict) positive definiteness while moving up and down in the ladder of dimensions by steps of one. These fractional operators are constructed to act appropriately on the Gegenbauer polynomials $\{C^\lambda_n\}$.

Key words: positive definite zonal functions; ultraspherical expansions; fractional integration; Gegenbauer polynomials.

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References

1. Abramowitz M., Stegun I.A. (Editors), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Dover Publications, New York, 1972.
2. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
3. Barbosa V.S., Menegatto V.A., Strictly positive definite kernels on two-point compact homogeneous spaces, Math. Inequal. Appl. 19 (2016), 743-756, arXiv:1505.00591.
4. Beatson R.K., zu Castell W., Dimension hopping and families of strictly positive definite radial functions on spheres, arXiv:1510.08658.
5. Billingsley P., Probability and measure, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York - Chichester - Brisbane, 1979.
6. Bingham N.H., Positive definite functions on spheres, Proc. Cambridge Philos. Soc. 73 (1973), 145-156.
7. Chen D., Menegatto V.A., Sun X., A necessary and sufficient condition for strictly positive definite functions on spheres, Proc. Amer. Math. Soc. 131 (2003), 2733-2740.
8. Chilès J.P., Delfiner P., Geostatistics. Modeling spatial uncertainty, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1999.
9. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. II, Bateman Manuscript Project, McGraw-Hill Book Co., New York, 1953.
10. Fasshauer G.E., Schumaker L.L., Scattered data fitting on the sphere, in Mathematical Methods for Curves and Surfaces, II (Lillehammer, 1997), Innov. Appl. Math., Vanderbilt University Press, Nashville, TN, 1998, 117-166.
11. Freeden W., Gervens T., Schreiner M., Constructive approximation on the sphere. With applications to geomathematics, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1998.
12. Gangolli R., Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy's Brownian motion of several parameters, Ann. Inst. H. Poincaré Sect. B (N.S.) 3 (1967), 121-226.
13. Gneiting T., Compactly supported correlation functions, J. Multivariate Anal. 83 (2002), 493-508.
14. Hardy G.H., Littlewood J.E., Some properties of fractional integrals. I, Math. Z. 27 (1928), 565-606.
15. Lévy P., Le mouvement Brownien fonction d'un point de la sphère de Riemann, Rend. Circ. Mat. Palermo 8 (1959), 297-310.
16. Matheron G., Les variables régionalisées et leur estimation: une application de la théorie des fonctions aléatoires aux sciences de la nature, Masson, Paris, 1965.
17. Matheron G., The intrinsic random functions and their applications, Adv. in Appl. Probab. 5 (1973), 439-468.
18. Menegatto V.A., Oliveira C.P., Peron A.P., Strictly positive definite kernels on subsets of the complex plane, Comput. Math. Appl. 51 (2006), 1233-1250.
19. NIST digital library of mathematical functions, available at http://dlmf.nist.gov/ .
20. Porcu E., Gregori P., Mateu J., La descente et la montée étendues [extended rises and descents]: the spatially $d$-anisotropic and the spatio-temporal case, Stoch. Environ. Res. Risk Assess. 21 (2007), 683-693.
21. Samko S.G., Kilbas A.A., Marichev O.I., Fractional integrals and derivatives. Theory and applications, Gordon and Breach Science Publishers, Yverdon, 1993.
22. Schaback R., Wu Z., Operators on radial functions, J. Comput. Appl. Math. 73 (1996), 257-270.
23. Schlather M., Construction of covariance functions and unconditional simulation of random fields, in Advances and Challenges in Space-Time Modelling of Natural Events, Lecture Notes in Statistics, Vol. 207, Editors E. Porcu, J. Montero, M. Schlather, Springer, Berlin, 2012, 25-54.
24. Schoenberg I.J., Positive definite functions on spheres, Duke Math. J. 9 (1942), 96-108.
25. Wendland H., Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math. 4 (1995), 389-396.
26. Wu Z.M., Compactly supported positive definite radial functions, Adv. Comput. Math. 4 (1995), 283-292.
27. zu Castell W., Recurrence relations for radial positive definite functions, J. Math. Anal. Appl. 271 (2002), 108-123.