How to Build Quasi-Interpolants: Application to Polyharmonic B-Splines

It is shown that B h may be defined by considering it as some regularisation of the Dirac distribution, and that it is better to determine B h by using some least square criterion than by using a “ ‘ P κ reproducing” criterion.

Abstract

We propose a quite general way to build “ quasi-interpolants ” on a cardinal grid, i.e. to determine some function B h such that the function has the general shape of the points ( jh,y j ) jɛZZ d . Different methods are proposed in order to get σ as close to the points ( jh, yj ) jɛZZd as wanted. It is shown that B h may be defined by considering it as some regularisation of the Dirac distribution, and that it is better to determine B h by using some least square criterion than by using a ‘ P κ reproducing” criterion. Special emphasis is given on “polyharmonic splines”: we define particular “m-harmonic splines” which are a natural generalisation of polynomial univariate B-splines. Extension is proposed for scatterred data.