It is well-known that non-linear approximation has an advantage over linear
schemes in the sense that it provides comparable approximation rates to those
of the linear schemes, but to a larger class of approximands. This was
established for spline approximations and for wavelet approximations, and more
recently for homogeneous radial basis function (surface spline) approximations.
However, no such results are known for the Gaussian function.

Problems involving approximation from scattered data where data is arranged
quasi-uniformly have been treated by RBF methods for decades. Treating data
with spatially varying density has not been investigated with the same
intensity, and is far less well understood. In this article we consider the
stability of surface spline interpolation (a popular type of RBF interpolation)
for data with nonuniform arrangements.

The purpose of this paper is to investigate RBF approximation with highly
nonuniform centers. Recently, DeVore and Ron have developed a notion of the
local density of a set of centers -- a notion that permits precise pointwise
error estimates for surface spline approximation. We give an equivalent,
alternative characterization of local density, one that allows effective
placement of centers at different resolutions.