Abstract

: Interpolation by translates of "radial" basis functions \Phi is optimal in the sense that it minimizes the pointwise error functional among all comparable quasi--interpolants on a certain "native" space of functions 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 from 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 . AMS Classification: 41A15, 41A25, 41A30, 41A63, 65D10 Keywords: Radial Basis Functions, Multivariate Approximation, Approximation Order 1 Introduction Given a continuous real--valued function \Phi on IR d and a nonnegative integer m, we consider approximations by finitely many translates \Phi(\Delta \Gamma x j ); 1 j N , of \Phi together with...

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