Approximation by radial basis functions with finitely many centers

Interpolation by translates of “radial” basis functions Φ is optimal in the sense that it minimizes the pointwise error functional among all comparable quasiinterpolants on a certain “native” space of functions $mathcal{F}_Phi $ . Since these spaces are rather small for cases where Φ is smooth, we study the behavior of interpolants on larger spaces of the form $mathcal{F}_{Phi _0 } $ for less smooth functions Φ₀. It turns out that interpolation by translates of Φ to mollifications of functionsf from $mathcal{F}_{Phi _0 } $ yields approximations tof that attain the same asymptotic error bounds as (optimal) interpolation off by translates of Φ₀ on $mathcal{F}_{Phi _0 } $ .