Approximation by Conditionally Positive Definite Functions with Finitely Many Centers

Abstract. The theory of interpolation by using conditionally positive definite function provides optimal error bounds when the basis function φ is smooth and the approximant f is in a certain native space Fφ. The space Fφ, however, is very small for the case where φ is smooth. Hence, in this study, we are interested in the approximation power of interpolation to mollifications of functions in Sobolev space. Specifically, it turns out that interpolation to mollifications provides spectral error bounds depending only on the smoothness of the functions f. In the last decades or so, there has been considerable progress concerning the scattered data approximation problem in two or more dimensions. In particular, the methods of radial basis function approximation are becoming increasingly popular. Usually, the starting point of the approximation proces