## Bernoulli

• Bernoulli
• Volume 19, Number 4 (2013), 1327-1349.

### Strictly and non-strictly positive definite functions on spheres

Tilmann Gneiting

#### Abstract

Isotropic positive definite functions on spheres play important roles in spatial statistics, where they occur as the correlation functions of homogeneous random fields and star-shaped random particles. In approximation theory, strictly positive definite functions serve as radial basis functions for interpolating scattered data on spherical domains. We review characterizations of positive definite functions on spheres in terms of Gegenbauer expansions and apply them to dimension walks, where monotonicity properties of the Gegenbauer coefficients guarantee positive definiteness in higher dimensions. Subject to a natural support condition, isotropic positive definite functions on the Euclidean space $\mathbb{R} ^{3}$, such as Askey’s and Wendland’s functions, allow for the direct substitution of the Euclidean distance by the great circle distance on a one-, two- or three-dimensional sphere, as opposed to the traditional approach, where the distances are transformed into each other. Completely monotone functions are positive definite on spheres of any dimension and provide rich parametric classes of such functions, including members of the powered exponential, Matérn, generalized Cauchy and Dagum families. The sine power family permits a continuous parameterization of the roughness of the sample paths of a Gaussian process. A collection of research problems provides challenges for future work in mathematical analysis, probability theory and spatial statistics.

#### Article information

Source
Bernoulli, Volume 19, Number 4 (2013), 1327-1349.

Dates
First available in Project Euclid: 27 August 2013

Permanent link to this document
https://projecteuclid.org/euclid.bj/1377612854

Digital Object Identifier
doi:10.3150/12-BEJSP06

Mathematical Reviews number (MathSciNet)
MR3102554

Zentralblatt MATH identifier
1283.62200

#### Citation

Gneiting, Tilmann. Strictly and non-strictly positive definite functions on spheres. Bernoulli 19 (2013), no. 4, 1327--1349. doi:10.3150/12-BEJSP06. https://projecteuclid.org/euclid.bj/1377612854

#### References

• Adler, R.J. (2009). The Geometry of Random Fields, SIAM Classics ed. Philadelphia: SIAM.
• Askey, R. (1973). Radial characteristic functions. Technical Report no. 1262, Mathematics Research Center, Univ. Wisconsin–Madison.
• Askey, R. and Fitch, J. (1969). Integral representations for Jacobi polynomials and some applications. J. Math. Anal. Appl. 26 411–437.
• Banerjee, S. (2005). On geodetic distance computations in spatial modeling. Biometrics 61 617–625.
• Beatson, R.K., zu Castell, W. and Xu, Y. (2011). A Pólya criterion for (strict) positive definiteness on the sphere. Preprint. Available at arXiv:1110.2437v1.
• Berg, C., Mateu, J. and Porcu, E. (2008). The Dagum family of isotropic correlation functions. Bernoulli 14 1134–1149.
• Bingham, N.H. (1973). Positive definite functions on spheres. Math. Proc. Cambridge Philos. Soc. 73 145–156.
• Cavoretto, R. and De Rossi, A. (2010). Fast and accurate interpolation of large scattered data sets on the sphere. J. Comput. Appl. Math. 234 1505–1521.
• Chen, D., Menegatto, V.A. and Sun, X. (2003). A necessary and sufficient condition for strictly positive definite functions on spheres. Proc. Amer. Math. Soc. 131 2733–2740.
• Daley, D.J. and Porcu, E. (2013). Dimension walks and Schoenberg spectral measures. Proc. Amer. Math. Soc. 141. To appear.
• Devinatz, A. (1959). On the extensions of positive definite functions. Acta Math. 102 109–134.
• Digital Library of Mathematical Functions (2011). Release 2011-07–01. Available at http://dlmf.nist.gov.
• Fasshauer, G.E. and Schumaker, L.L. (1998). Scattered data fitting on the sphere. In Mathematical Methods for Curves and Surfaces, II (Lillehammer, 1997) (M. Daehlen, T. Lyche and L.L. Schumaker, eds.). Innov. Appl. Math. 117–166. Nashville, TN: Vanderbilt Univ. Press.
• Furrer, R., Genton, M.G. and Nychka, D. (2006). Covariance tapering for interpolation of large spatial datasets. J. Comput. Graph. Statist. 15 502–523.
• Gaspari, G. and Cohn, S.E. (1999). Construction of correlation functions in two and three dimensions. Q. J. Roy. Meteorol. Soc. 125 723–757.
• Gneiting, T. (1998a). On $\alpha$-symmetric multivariate characteristic functions. J. Multivariate Anal. 64 131–147.
• Gneiting, T. (1998b). Simple tests for the validity of correlation function models on the circle. Statist. Probab. Lett. 39 119–122.
• Gneiting, T. (1999a). Correlation functions for atmospheric data analysis. Q. J. Roy. Meteorol. Soc. 125 2449–2464.
• Gneiting, T. (1999b). Radial positive definite functions generated by Euclid’s hat. J. Multivariate Anal. 69 88–119.
• Gneiting, T. (2001). Criteria of Pólya type for radial positive definite functions. Proc. Amer. Math. Soc. 129 2309–2318.
• Gneiting, T. (2002). Compactly supported correlation functions. J. Multivariate Anal. 83 493–508.
• Gneiting, T. (2013). Supplement to “Strictly and non-strictly positive definite functions on spheres.” DOI:10.3150/12-BEJSP06SUPP.
• Gneiting, T. and Schlather, M. (2004). Stochastic models that separate fractal dimension and the Hurst effect. SIAM Rev. 46 269–282.
• Guttorp, P. and Gneiting, T. (2006). Studies in the history of probability and statistics. XLIX. On the Matérn correlation family. Biometrika 93 989–995.
• Hamill, T.M., Whitaker, J.S. and Snyder, C. (2001). Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Mon. Wea. Rev. 129 2776–2790.
• Hansen, L.V., Thorarinsdottir, T.L. and Gneiting, T. (2011). Lévy particles: Modelling and simulating star-shaped random sets. Research Report 2011/04, Centre for Stochastic Geometry and Advanced Bioimaging, Univ. Aarhus. Available at http://data.imf.au.dk/publications/csgb/2011/imf-csgb-2011-04.pdf.
• Hjorth, P., Lison\uek, P., Markvorsen, S. and Thomassen, C. (1998). Finite metric spaces of strictly negative type. Linear Algebra Appl. 270 255–273.
• Huang, C., Zhang, H. and Robeson, S.M. (2011). On the validity of commonly used covariance and variogram functions on the sphere. Math. Geosci. 43 721–733.
• Jones, R.H. (1963). Stochastic processes on a sphere. Ann. Math. Statist. 34 213–218.
• Le Gia, Q.T., Sloan, I.H. and Wendland, H. (2010). Multiscale analysis in Sobolev spaces on the sphere. SIAM J. Numer. Anal. 48 2065–2090.
• Lévy, P. (1961). Quelques problèmes non résolus de la théorie des fonctions caractéristiques. Ann. Mat. Pura Appl. (4) 53 315–331.
• Menegatto, V.A. (1994). Strictly positive definite kernels on the Hilbert sphere. Appl. Anal. 55 91–101.
• Menegatto, V.A. (1995). Strictly positive definite kernels on the circle. Rocky Mountain J. Math. 25 1149–1163.
• Menegatto, V.A., Oliveira, C.P. and Peron, A.P. (2006). Strictly positive definite kernels on subsets of the complex plane. Comput. Math. Appl. 51 1233–1250.
• Miller, K.S. and Samko, S.G. (2001). Completely monotonic functions. Integral Transforms Spec. Funct. 12 389–402.
• Narcowich, F.J. (1995). Generalized Hermite interpolation and positive definite kernels on a Riemannian manifold. J. Math. Anal. Appl. 190 165–193.
• Narcowich, F.J. and Ward, J.D. (2002). Scattered data interpolation on spheres: Error estimates and locally supported basis functions. SIAM J. Math. Anal. 33 1393–1410.
• Paley, R.E.A.C. and Wiener, N. (1934). Fourier Transforms in the Complex Domain. New York: Amer. Math. Soc.
• Sasvári, Z. (1994). Positive Definite and Definitizable Functions. Mathematical Topics 2. Berlin: Akademie Verlag.
• Schoenberg, I.J. (1938). Metric spaces and completely monotone functions. Ann. of Math. (2) 39 811–841.
• Schoenberg, I.J. (1942). Positive definite functions on spheres. Duke Math. J. 9 96–108.
• Schreiner, M. (1997). Locally supported kernels for spherical spline interpolation. J. Approx. Theory 89 172–194.
• Soubeyrand, S., Enjalbert, J. and Sache, I. (2008). Accounting for roughness of circular processes: Using Gaussian random processes to model the anisotropic spread of airborne plant disease. Theoret. Popul. Biol. 73 92–103.
• Sun, X. (1993). Conditionally positive definite functions and their application to multivariate interpolations. J. Approx. Theory 74 159–180.
• Wendland, H. (1995). Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4 389–396.
• Wood, A.T.A. (1995). When is a truncated covariance function on the line a covariance function on the circle? Statist. Probab. Lett. 24 157–164.
• Xu, Y. and Cheney, E.W. (1992). Strictly positive definite functions on spheres. Proc. Amer. Math. Soc. 116 977–981.
• Yadrenko, M.Ĭ. (1983). Spectral Theory of Random Fields. Translation Series in Mathematics and Engineering. New York: Optimization Software.
• Yaglom, A.M. (1987). Correlation Theory of Stationary and Related Random Functions. Vol. I: Basic Results. Springer Series in Statistics. New York: Springer.
• Ziegel, J. (2013). Convolution roots and differentiability of isotropic positive definite functions on spheres. Proc. Amer. Math. Soc. 141. To appear.

#### Supplemental materials

• Supplementary material: Supplement to “Strictly and non-strictly positive definite functions on spheres”. Appendix A states and proves further criteria of Pólya type, thereby complementing Section 4.2. Appendix B studies an example that involves oscillating trigonometric and Bessel functions, as hinted at in Section 4.5. Appendix C describes open problems that aim to stimulate future research in areas ranging from harmonic analysis to spatial statistics.