Approximation Properties of Sobolev Splines and the Construction of Compactly Supported Equivalents

In this paper, we construct compactly supported radial basis functions that satisfy optimal approximation properties. Error estimates are determined by relating these basis functions to the class of Sobolev splines. Furthermore, we derive new rates for approximation by linear combinations of nonuniform translates of the Sobolev splines. Our results extend previous work as we obtain rates for basis functions of noninteger order, and we address approximation with respect to the $L^{\infty}$ norm. We also use bandlimited approximation to determine rates for target functions with lower order smoothness.