Abstract
This paper is concerned with the study of continuous least squares approximation on a bounded domain in $\mathbb{R}^s $ by certain classes of radial functions. The approximating subspace is spanned by translates $F( \cdot - x_j )$ of a given radial function F, where the (distinct) “centers” $\{ x_j \} _{j = 1}^N $ are allowed to be scattered. The main result gives quantitative estimates for the Euclidean norms of the inverses of these least squares matrices. In general, the estimates involve the dimension of the ambient space, the minimal separation distance between the centers, the number of centers, and of course the function itself. However, if F is the scaled Gaussian, it is possible to dispense with the dependence on the number of centers. Also established along the way are results involving radial interpolation matrices where the interpolation points are small perturbations of the centers. These results are perhaps of independent interest as previous interpolation results had been obtained only for interpolation at the centers.
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