A general multipurpose interpolation procedure: the magic points

Abstract

Lagrangian interpolation is a classical way to approximate generalfunctions by finite sums of well chosen, pre-defined, linearlyindependent interpolating functions; it is much simpler to implement thandetermining the best fits with respect to some Banach (or even Hilbert)norms. In addition, only partial knowledge is required (here values on someset of points). The problem of defining the best sample of points isnevertheless rather complex and is in general open. In this paper wepropose a way to derive such sets of points. We do not claim that thepoints resulting from the construction explained here are optimal in anysense. Nevertheless, the resulting interpolation method is proven to work under certain hypothesis, theprocess is very general and simple to implement, and compared to situationswhere the best behavior is known, it is relatively competitive.