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Strictly positive definite functions on spheres in Euclidean spaces

Authors: Amos Ron and Xingping Sun
Journal: Math. Comp. 65 (1996), 1513-1530
MSC (1991): Primary 42A82, 41A05; Secondary 33C55, 33C90
MathSciNet review: 1370856
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Abstract: In this paper we study strictly positive definite functions on the unit sphere of the $m$-dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on spheres. Since positive definite functions on the sphere were already characterized by Schoenberg some fifty years ago, the issue here is to determine what kind of positive definite functions are actually strictly positive definite. The study of this problem was initiated recently by Xu and Cheney (Proc. Amer. Math. Soc. 116 (1992), 977–981), where certain sufficient conditions were derived. A new approach, which is based on a critical connection between this problem and that of multivariate polynomial interpolation on spheres, is presented here. The relevant interpolation problem is subsequently analyzed by three different complementary methods. The first is based on the de Boor-Ron general “least solution for the multivariate polynomial interpolation problem”. The second, which is suitable only for $m=2$, is based on the connection between bivariate harmonic polynomials and univariate analytic polynomials, and reduces the problem to the structure of the integer zeros of bounded univariate exponentials. Finally, the last method invokes the realization of harmonic polynomials as the polynomial kernel of the Laplacian, thereby exploiting some basic relations between homogeneous ideals and their polynomial kernels.

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Additional Information

Amos Ron
Affiliation: Department of Computer Science, University of Wisconsin-Madison, Madison, Wisconsin 53706

Xingping Sun
Affiliation: Department of Mathematics, Southwest Missouri State University, Springfield, Missouri 65804

Received by editor(s): February 7, 1994
Received by editor(s) in revised form: February 22, 1995, and July 5, 1995
Article copyright: © Copyright 1996 American Mathematical Society