Approximation by radial basis functions with finitely many centers

Abstract

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$\mathcal{F}_{\Phi _0 }$.