We consider error estimates for the interpolation by a special class of compactly supported radial basis functions. These functions consist of a univariate polynomial within their support and are of minimal degree depending on space dimension and smoothness. Their associated "native" Hilbert spaces are shown to be norm-equivalent to Sobolev spaces. Thus we can derive approximation orders for functions from Sobolev spaces which are comparable to those of thin-plate-spline interpolation. Finally, we investigate the numerical stability of the interpolation process.

. A general theory is provided that allows to write multivariate Fourier transforms or convolutions of radial functions as very simple univariate operations. As a byproduct, an interesting group of operators fI ff g ff2IR with I ff+fi = I ff ffi I fi = I fi ffi I ff is defined. It contains the classical derivatives as I \Gamma1 = d dr and is intimately connected to the Fourier transform. Applications to the construction of new positive definite radial functions and to new identities for special functions are included. x1. Radial functions Among all functions g : IR d n f0g ! IR we consider those which can be written as a univariate function of the Euclidean norm kxk 2 on IR d . Definition 1.1. A function g : IR d nf0g ! IR is radial if there is a function f : IR?0 ! IR such that g(x) = f(kxk 2 2 =2); x 2 IR d n f0g: (1:1) Remark 1.2. Functions defined on all of IR d are treated similarly. Note that we do not use g(x) = f(kxk 2 ) for reasons that will soon be apparent. ...