A unified view of space-time covariance functions through Gelfand pairs. (English) Zbl 1465.43002
This paper gives a unifying framework that encompasses all the relevant contributions in the construction and characterization of space-time covariance functions. More precisely, positive definite integrable functions on a product of two Gelfand pairs as an integral of positive definite functions on one of the Gelfand pairs with respect to the Plancherel measure on the dual of the other Gelfand pair are characterized. In the very special case where the Gelfand pairs are Euclidean groups and the compact subgroups are reduced to the identity, the characterization is a much cited result in spatio-temporal statistics due to Cressie, Huang and Gneiting. When one of the Gelfand pairs is compact, the characterization leads to results about expansions in spherical functions with positive definite expansion functions, thereby recovering recent results of the author in collaboration with Peron and Porcu. In the special case where the compact Gelfand pair consists of orthogonal groups, the characterization is important in geostatistics and covers a recent result of Porcu and White.
Reviewer: Elijah Liflyand (Ramat-Gan)
MSC:
| 43A25 | Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups |
| 43A35 | Positive definite functions on groups, semigroups, etc. |
| 43A75 | Harmonic analysis on specific compact groups |
| 43A90 | Harmonic analysis and spherical functions |
Keywords:
positive definite functions; covariance functions; harmonic analysis on Gelfand pairs; Plancherel measure; spherical functionReferences:
| [1] | Andrews, GE; Askey, R.; Roy, R., Special Functions (1999), Cambridge: Cambridge University Press, Cambridge · Zbl 0920.33001 · doi:10.1017/CBO9781107325937 |
| [2] | Apanasovich, TV; Genton, MG, Cross-covariance functions for multivariate random fields based on latent dimensions, Biometrika, 97, 15-30 (2010) · Zbl 1183.62164 · doi:10.1093/biomet/asp078 |
| [3] | Berg, C., Dirichlet forms on symmetric spaces, Ann. Inst. Fourier Grenoble, 23, 1, 135-156 (1973) · Zbl 0243.31013 · doi:10.5802/aif.448 |
| [4] | Berg, C.; Forst, G., Potential Theory on Locally Compact Abelian Groups (1975), Berlin: Springer, Berlin · Zbl 0308.31001 · doi:10.1007/978-3-642-66128-0 |
| [5] | Berg, C.; Porcu, E., From Schoenberg coefficients to Schoenberg functions, Constr. Approx., 45, 217-241 (2017) · Zbl 1362.43004 · doi:10.1007/s00365-016-9323-9 |
| [6] | Berg, C.; Peron, AP; Porcu, E., Orthogonal expansions related to compact Gelfand pairs, Expo. Math., 36, 259-277 (2018) · Zbl 1407.43002 · doi:10.1016/j.exmath.2017.10.005 |
| [7] | Charalambides, CA, Enumerative Combinatorics (2002), Boca Raton: Chapman & Hall, Boca Raton · Zbl 1001.05001 |
| [8] | Cooper, JLB, Positive definite functions of a real variable, Proc. Lond. Math. Soc., 10, 53-66 (1960) · Zbl 0099.31102 · doi:10.1112/plms/s3-10.1.53 |
| [9] | Cressie, N.; Huang, H-C, Classes of nonseparable, spatiotemporal stationary covariance functions, J. Am. Stat. Assoc., 94, 1330-1340 (1999) · Zbl 0999.62073 · doi:10.1080/01621459.1999.10473885 |
| [10] | Dai, F.; Xu, Y., Approximation Theory and Harmonic Analysis on Spheres and Balls (2013), Berlin: Springer, Berlin · Zbl 1275.42001 · doi:10.1007/978-1-4614-6660-4 |
| [11] | Dawson, M.; Ólafsson, G.; Wolf, JA, Direct systems of spherical functions and representations, J. Lie Theory, 23, 711-729 (2013) · Zbl 1280.43003 |
| [12] | Dieudonné, J., Éléments D’Analyse, Tome VI. Chapitre XXII (1975), Paris: Gauthier-Villars, Paris · Zbl 0303.43001 |
| [13] | Dixmier, J., Les \(C^*\)-algèbres et Leurs Représentations (1964), Paris: Gauthier-Villars, Paris · Zbl 0152.32902 |
| [14] | Faraut, J., Analysis on Lie Groups. An Introduction (2008), Cambridge: Cambridge University Press, Cambridge · Zbl 1147.22001 · doi:10.1017/CBO9780511755170 |
| [15] | Furrer, R.; Porcu, E.; Nychka, D., A 30 years of space-time covariance functions, WIREs Comput. Stat. (2020) · Zbl 07910735 · doi:10.1002/wics.1512 |
| [16] | Gneiting, T., Nonseparable, stationary covariance functions for space-time data, J. Am. Stat. Assoc., 97, 590-600 (2002) · Zbl 1073.62593 · doi:10.1198/016214502760047113 |
| [17] | Gneiting, T., Strictly and non-strictly positive definite functions on spheres, Bernoulli, 19, 4, 1327-1349 (2013) · Zbl 1283.62200 · doi:10.3150/12-BEJSP06 |
| [18] | Kotz, S.; Ostrovskii, IV; Hayfavi, A., Analytic and asymptotic properties of Linnik’s probability densities. I., J. Math. Anal. Appl., 193, 353-371 (1995) · Zbl 0831.60020 · doi:10.1006/jmaa.1995.1240 |
| [19] | Linnik, Ju.V.: Linear forms and statistical criteria, I, II. Select. Transl. Math. Stat. Probab. 3, 1-90 (1963). [Original paper appeared in Ukrainskii Mat. Zh. 5 (1953), 207-290.] · Zbl 0052.36701 |
| [20] | Mateu, J., Porcu, E.: Positive Definite Functions: From Schoenberg to Space-Time Challenges. University Jaume I, Castellon, Spain, Department of Mathematics (2008) |
| [21] | Phillips, TRL; Schmidt, KM, On unbounded positive definite functions, Math. Pannon., 26, 2, 33-51 (2018) · Zbl 1474.42030 |
| [22] | Porcu, E.; White, P., Towards a complete picture of stationary covariance functions on spheres cross time, Electron. J. Stat., 13, 2566-2594 (2019) · Zbl 1426.62285 · doi:10.1214/19-EJS1593 |
| [23] | Porcu, E., Cleanthous, G., Georgiadis, G., White, P.A., Alegria A.: Random fields on the hypertorus: Covariance modeling, regularities and approximations. Technical Report: Submitted for publication |
| [24] | Porcu, E.; Gregori, P.; Mateu, J., Nonseparable stationary anisotropic space-time covariance functions, Stoch. Environ. Res. Risk Assess., 21, 113-122 (2006) · doi:10.1007/s00477-006-0048-3 |
| [25] | Rudin, W., Fourier Analysis on Groups (1962), New York: Interscience, New York · Zbl 0107.09603 |
| [26] | Sasvári, Z., Positive Definite and Definitizable Functions (1994), Berlin: Akademie, Berlin · Zbl 0815.43003 |
| [27] | Schoenberg, IJ, Positive definite functions on spheres, Duke Math. J., 9, 96-108 (1942) · Zbl 0063.06808 · doi:10.1215/S0012-7094-42-00908-6 |
| [28] | Stein, EM; Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces (1971), Princeton: Princeton University Press, Princeton · Zbl 0232.42007 |
| [29] | van Dijk, G., Introduction to Harmonic Analysis and Generalized Gelfand Pairs De Gruyter Studies in Mathematics, 36 (2009), Berlin: Walter de Gruyter & Co, Berlin · Zbl 1184.43001 · doi:10.1515/9783110220209 |
| [30] | Widder, DV, The Laplace Transform (1941), Princeton: Princeton University Press, Princeton · JFM 67.0384.01 |
| [31] | Wolf, JA, Harmonic Analysis on Commutative Spaces (2007), New York: American Mathematical Society, New York · Zbl 1156.22010 · doi:10.1090/surv/142 |
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