Abstract
Let \(S^m\) denote the unit sphere in \(R^{m + 1}\) and \(d_m\) the geodesic distance in \(S^m\). A spherical‐basis function approximant is a function of the form \(s\left( x \right) = \sum\nolimits_{j = 1}^M {a_j \varphi \left( {d_m \left( {x,x_j } \right)} \right)\;x \in S^m }\), where \(\left( {a_j } \right)_1^M\) are real constants, \(\varphi :\left[ {0,{\pi }} \right] \to R\) is a fixed function, and \(\left( {x_j } \right)_1^M\) is a set of distinct points in \(S^m\). It is known that if \(\varphi\) is a strictly positive definite function in \(S^m\), then the interpolation matrix \(\left( {\varphi \left( {d_m \left( {x_j ,x_k } \right)} \right)} \right)_{j,k = 1}^M\) is positive definite, hence invertible, for every choice of distinct points \(\left( {x_j } \right)_1^M\) and every positive integer M. The paper studies a salient subclass of such functions \(\varphi\), and provides stability estimates for the associated interpolation matrices.
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Narcowich, F., Sivakumar, N. & Ward, J. Stability results for scattered‐data interpolation on Euclidean spheres. Advances in Computational Mathematics 8, 137–163 (1998). https://doi.org/10.1023/A:1018996230401
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DOI: https://doi.org/10.1023/A:1018996230401