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An extension of a theorem of Schoenberg to products of spheres. (English) Zbl 1348.43008

Isotropy and positive definiteness for kernels on single sphere were considered by I. J. Schoenberg [Duke Math. J. 9, 96–108 (1942; Zbl 0063.06808)]. The authors extend Schoenberg’s theorem to a product of spheres \(S^{m}\times S^{M}\), when \(m,M\) are both finite and also in the case where either \(m=\infty\) or \(M=\infty\).

MSC:

43A35 Positive definite functions on groups, semigroups, etc.
33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
33C55 Spherical harmonics
42A10 Trigonometric approximation
42A82 Positive definite functions in one variable harmonic analysis

Citations:

Zbl 0063.06808
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References:

[1] R. Alexander and K. B. Stolarsky, Extremal problems of distance geometry related to energy integrals , Trans. Amer. Math. Soc. 193 (1974), 1-31. · Zbl 0293.52005 · doi:10.2307/1996898
[2] C. Bachoc, Semidefinite programming, harmonic analysis and coding theory , preprint, [cs.IT]. arXiv:0909.4767v2
[3] C. Bachoc, Y. Ben-Haim, and S. Litsyn, Bounds for codes in products of spaces, Grassmann, and Stiefel manifolds , IEEE Trans. Inform. Theory 54 (2008), no. 3, 1024-1035. · Zbl 1311.94116 · doi:10.1109/TIT.2007.915916
[4] C. Berg, J. P. R. Christensen, and P. Ressel, Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions , Grad. Texts in Math. 100 , Springer, New York, 1984. · Zbl 0619.43001
[5] C. Berg and E. Porcu, From Schoenberg coefficients to Schoenberg functions , preprint, to appear in Constr. Approx., [math.CA]. arXiv:1505.05682v2 · Zbl 1362.43004
[6] S. Bochner, Harmonic Analysis and the Theory of Probability , Univ. of California Press, Berkeley and Los Angeles, 1955. · Zbl 0068.11702
[7] D. Chen, V. A. Menegatto, and X. Sun, A necessary and sufficient condition for strictly positive definite functions on spheres , Proc. Amer. Math. Soc. 131 (2003), no. 9, 2733-2740. · Zbl 1125.43300 · doi:10.1090/S0002-9939-03-06730-3
[8] E. W. Cheney, “Approximation using positive definite functions” in Approximation Theory VIII, Vol. 1 (College Station, TX, 1995) , Ser. Approx. Decompos. 6 , World Sci. Publ., River Edge, NJ, 1995, 145-168. · Zbl 1137.41338
[9] F. Dai and Y. Xu, Approximation Theory and Harmonic Analysis on Spheres and Balls , Springer Monogr. Math., Springer, New York, 2013. · Zbl 1275.42001 · doi:10.1007/978-1-4614-6660-4
[10] W. F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation , Grundlehren Math. Wiss. 207 , Springer, New York, 1974. · Zbl 0278.30004
[11] S. R. Ghorpade and B. V. Limaye, A Course in Multivariable Calculus and Analysis , Undergrad. Texts Math., Springer, New York, 2010. · Zbl 1186.26001 · doi:10.1007/978-1-4419-1621-1
[12] T. Gneiting, Strictly and non-strictly positive definite functions on spheres , Bernoulli 19 (2013), no. 4, 1327-1349. · Zbl 1283.62200 · doi:10.3150/12-BEJSP06
[13] R. A. Horn and C. R. Johnson, Matrix Analysis , 2nd ed., Cambridge Univ. Press, Cambridge, 2013. · Zbl 1267.15001
[14] V. A. Menegatto, Strictly positive definite kernels on the Hilbert sphere , Appl. Anal. 55 (1994), no. 1-2, 91-101. · Zbl 0873.41005 · doi:10.1080/00036819408840292
[15] V. A. Menegatto, Strict positive definiteness on spheres , Analysis (Munich) 19 (1999), no. 3, 217-233. · Zbl 0978.42013 · doi:10.1524/anly.1999.19.3.217
[16] 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. · Zbl 1153.41307 · doi:10.1016/j.camwa.2006.04.006
[17] K. S. Miller and S. G. Samko, Completely monotonic functions , Integral Transforms Spec. Funct. 12 (2001), no. 4, 389-402. · Zbl 1035.26012 · doi:10.1080/10652460108819360
[18] C. Müller, Analysis of Spherical Symmetries in Euclidean Spaces , Appl. Math. Sci. 129 , Springer, New York, 1998.
[19] A. Ron and X. Sun, Strictly positive definite functions on spheres in Euclidean spaces , Math. Comp. 65 (1996), no. 216, 1513-1530. · Zbl 0853.42018 · doi:10.1090/S0025-5718-96-00780-6
[20] I. J. Schoenberg, Positive definite functions on spheres , Duke Math. J. 9 (1942), 96-108. · Zbl 0063.06808 · doi:10.1215/S0012-7094-42-00908-6
[21] G. Szegö, Orthogonal Polynomials , 4th ed., Amer. Math. Soc. Colloq. Publ. XXIII , Amer. Math. Soc., Providence, 1975.
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