Strictly and non-strictly positive definite functions on spheres

Bernoulli

Volumn 19, Issue 4, 2013, Pages 1327-1349

  Gneiting, Tilmann ^a  

^a UNIVERSITY OF HEIDELBERG   (Germany)

Author keywords

Completely monotone; Covariance localization; Fractal index; Interpolation of scattered data; Isotropic; Locally supported; Multiquadric; P lya criterion; Radial basis function; Schoenberg coefficients

Indexed keywords

EID: 84879548201     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/12-BEJSP06     Document Type: Article

References (50)

1. Adler, R.J. (2009). The Geometry of Random Fields, SIAM Classics ed. Philadelphia: SIAM.
2. Askey, R. (1973). Radial characteristic functions. Technical Report no. 1262, Mathematics Research Center, Univ. Wisconsin-Madison.
3. Askey, R. and Fitch, J. (1969). Integral representations for Jacobi polynomials and some applications. J. Math. Anal. Appl. 26 411-437. MR0237847
4. Banerjee, S. (2005). On geodetic distance computations in spatial modeling. Biometrics 61 617-625. MR2140936
5. 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.
6. Berg, C., Mateu, J. and Porcu, E. (2008). The Dagum family of isotropic correlation functions. Bernoulli 14 1134-1149. MR2543589
7. Bingham, N.H. (1973). Positive definite functions on spheres. Math. Proc. Cambridge Philos. Soc. 73 145-156. MR0339308
8. 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. MR2610367
9. 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. MR1974330
10. Daley, D.J. and Porcu, E. (2013). Dimension walks and Schoenberg spectral measures. Proc. Amer. Math. Soc. 141. To appear.
11. Devinatz, A. (1959). On the extensions of positive definite functions. Acta Math. 102 109-134. MR0109992
12. Digital Library of Mathematical Functions (2011). Release 2011-07-01. Available at http://dlmf.nist.gov.
13. 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. MR1640548
14. Furrer, R., Genton, M.G. and Nychka, D. (2006). Covariance tapering for interpolation of large spatial datasets. J. Comput. Graph. Statist. 15 502-523. MR2291261
15. Gaspari, G. and Cohn, S.E. (1999). Construction of correlation functions in two and three dimensions. Q. J. Roy. Meteorol. Soc. 125 723-757.
16. Gneiting, T. (1998a). On α-symmetric multivariate characteristic functions. J. Multivariate Anal. 64 131-147. MR1621926
17. Gneiting, T. (1998b). Simple tests for the validity of correlation function models on the circle. Statist. Probab. Lett. 39 119-122. MR1652540
18. Gneiting, T. (1999a). Correlation functions for atmospheric data analysis. Q. J. Roy. Meteorol. Soc. 125 2449-2464.
19. Gneiting, T. (1999b). Radial positive definite functions generated by Euclid's hat. J. Multivariate Anal. 69 88-119. MR1701408
20. Gneiting, T. (2001). Criteria of Pólya type for radial positive definite functions. Proc. Amer. Math. Soc. 129 2309-2318. MR1823914
21. Gneiting, T. (2002). Compactly supported correlation functions. J. Multivariate Anal. 83 493-508. MR1945966
22. Gneiting, T. (2013). Supplement to "Strictly and non-strictly positive definite functions on spheres." DOI:10.3150/12-BEJSP06SUPP.
23. Gneiting, T. and Schlather, M. (2004). Stochastic models that separate fractal dimension and the Hurst effect. SIAM Rev. 46 269-282. MR2114455
24. Guttorp, P. and Gneiting, T. (2006). Studies in the history of probability and statistics. XLIX. On the Matérn correlation family. Biometrika 93 989-995. MR2285084
25. 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.
26. 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.
27. Hjorth, P., Lisoněk, P., Markvorsen, S. and Thomassen, C. (1998). Finite metric spaces of strictly negative type. Linear Algebra Appl. 270 255-273. MR1484084
28. 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. MR2824128
29. Jones, R.H. (1963). Stochastic processes on a sphere. Ann. Math. Statist. 34 213-218. MR0170378
30. 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. MR2740542
31. Lévy, P. (1961). Quelques problèmes non résolus de la théorie des fonctions caractéristiques. Ann. Mat. Pura Appl. (4) 53 315-331. MR0125414
32. Menegatto, V.A. (1994). Strictly positive definite kernels on the Hilbert sphere. Appl. Anal. 55 91-101. MR1379646
33. Menegatto, V.A. (1995). Strictly positive definite kernels on the circle. Rocky Mountain J. Math. 25 1149-1163. MR1357116
34. 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. MR2235825
35. Miller, K.S. and Samko, S.G. (2001). Completely monotonic functions. Integral Transforms Spec. Funct. 12 389-402. MR1872377
36. Narcowich, F.J. (1995). Generalized Hermite interpolation and positive definite kernels on a Riemannian manifold. J. Math. Anal. Appl. 190 165-193. MR1314111
37. 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. MR1920637
38. Paley, R.E.A.C. and Wiener, N. (1934). Fourier Transforms in the Complex Domain. New York: Amer. Math. Soc.
39. Sasvári, Z. (1994). Positive Definite and Definitizable Functions. Mathematical Topics 2. Berlin: Akademie Verlag. MR1270018
40. Schoenberg, I.J. (1938). Metric spaces and completely monotone functions. Ann. of Math. (2) 39 811-841. MR1503439
41. Schoenberg, I.J. (1942). Positive definite functions on spheres. Duke Math. J. 9 96-108. MR0005922
42. Schreiner, M. (1997). Locally supported kernels for spherical spline interpolation. J. Approx. Theory 89 172-194. MR1447837
43. 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.
44. Sun, X. (1993). Conditionally positive definite functions and their application to multivariate interpolations. J. Approx. Theory 74 159-180. MR1226354
45. Wendland, H. (1995). Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4 389-396. MR1366510
46. 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. MR1354370
47. Xu, Y. and Cheney, E.W. (1992). Strictly positive definite functions on spheres. Proc. Amer. Math. Soc. 116 977-981. MR1096214
48. Yadrenko, M. Ǐ. (1983). Spectral Theory of Random Fields. Translation Series in Mathematics and Engineering. New York: Optimization Software. MR0697386
49. Yaglom, A.M. (1987). Correlation Theory of Stationary and Related Random Functions. Vol. I: Basic Results. Springer Series in Statistics. New York: Springer. MR0893393
50. Ziegel, J. (2013). Convolution roots and differentiability of isotropic positive definite functions on spheres. Proc. Amer. Math. Soc. 141. To appear.