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On a new condition for strictly
positive definite functions on spheres


Author: Michael Schreiner
Journal: Proc. Amer. Math. Soc. 125 (1997), 531-539
MSC (1991): Primary 43A35, 43A90, 42A82, 41A05
DOI: https://doi.org/10.1090/S0002-9939-97-03634-4
MathSciNet review: 1353398
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Abstract | References | Similar Articles | Additional Information

Abstract: Recently, Xu and Cheney (1992) have proved that if all the Legendre coefficients of a zonal function defined on a sphere are positive then the function is strictly positive definite. It will be shown in this paper that, even if finitely many of the Legendre coefficients are zero, the strict positive definiteness can be assured. The results are based on approximation properties of singular integrals, and provide also a completely different proof of the results of Xu and Cheney.


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  • 1. N. Aronszajn, Theory of Reproducing Kernels, Trans. Amer. Math. Soc., 68 (1950), 337-404 MR 14:479c
  • 2. A.P. Calderon and A. Zygmund, On a Problem of Mihlin, Trans. Amer. Math. Soc., 78 (1955), 209-224 MR 16:816f
  • 3. P.J. Davis, Interpolation and Approximation, Blaisdell Publishing Company, New York, Toronto, London, 1963 MR 28:393
  • 4. W. Freeden, T. Gervens, M. Schreiner, Tensor Spherical Harmonics and Tensor Spherical Splines, Manuscr. Geod. 19 (1994), 70-100
  • 5. T. Gronwall, On the Degree of Convergence of Laplace Series, Trans. Amer. Math. Soc., 15 (1914), 1-30
  • 6. C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, 17, Springer, Berlin, Heidelberg, New York, 1966 MR 33:7593
  • 7. R. Rummel, Satellite Gradiometry, Lecture Notes in Earth Sciences 7, Mathematical and Numerical Techniques in Physical Geodesy (H. Sünkel, ed), Springer, Berlin, 1986, 318-363 MR 88k:86004
  • 8. I.J. Schoenberg, Positive Definite Functions on Spheres, Duke Math. J., 9 (1942), 96-108 MR 3:232c
  • 9. M. Schreiner, Tensor Spherical Harmonics and Their Application in Satellite Gradiometry, PhD-Thesis, University of Kaiserslautern, Geomathematics Group, 1994
  • 10. G. Szegö, Orthogonal Polynomials, 2nd ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1959 MR 21:5029
  • 11. Y. Xu and E.W. Cheney, Strictly Positive Definite Functions On Spheres, Proc. Amer. Math. Soc., 116 (1992), 977-981 MR 93b:43005

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Additional Information

Michael Schreiner
Affiliation: University of Kaiserslautern, Laboratory of Technomathematics, Geomathematics Group, P.O. Box 30 49, 67653 Kaiserslautern, Germany
Email: schreiner@mathematik.uni-kl.de

DOI: https://doi.org/10.1090/S0002-9939-97-03634-4
Keywords: Positive definite functions, spherical interpolation
Received by editor(s): August 29, 1995
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society

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