Dual bases and discrete reproducing kernels: a unified framework for RBF and MLS approximation

Abstract

Moving least squares (MLS) and radial basis function (RBF) methods play a central role in multivariate approximation theory. In this paper we provide a unified framework for both RBF and MLS approximation. This framework turns out to be a linearly constrained quadratic minimization problem. We show that RBF approximation can be considered as a special case of MLS approximation. This sheds new light on both MLS and RBF approximation. Among the new insights are dual bases for the approximation spaces and certain discrete reproducing kernels.