APPROXIMATION BY RADIAL BASIS FUNCTIONS WITH FINITELY MANY CENTERS

Interpolation by translates of ''radial'' basis functions Phi is optim al in the sense that it minimizes the pointwise error functional among all comparable quasi-interpolants on a certain ''native'' space of fu nctions F-Phi. Since these spaces are rather small for cases where Phi is smooth, we study the behavior of interpolants on larger spaces of the form F-Phi 0, for less smooth functions Phi(0). It turns out that interpolation by translates of Phi to mollifications of functions f fr om F-Phi 0, yields approximations to f that attain the same asymptotic error bounds as (optimal) interpolation of f by translates of Phi(0) on F-Phi 0.