Stability results for scattered‐data interpolation on Euclidean spheres

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.