Approximation in Lp(Rd) from Spaces Spanned by the Perturbed Integer Translates of a Radial Function Original Research Article

The problem of approximating smooth Lp-functions from spaces spanned by the integer translates of a radially symmetric function φ is very well understood. In case the points of translation, Ξ, are scattered throughout Rd, the approximation problem is only well understood in the “stationary” setting. In this work, we provide lower bounds on the obtainable approximation orders in the “non-stationary” setting under the assumption that Ξ is a small perturbation of Zd. The functions which we can approximate belong to certain Besov spaces. Our results, which are similar in many respects to the known results for the case Ξ=Zd, apply specifically to the examples of the Gauss kernel and the generalized multiquadric.