AN INTRODUCTION TO SCHOENBERG APPROXIMATION

For a given function B and a non-zero real number h, Schoenberg's approximation defines from some data (jh, y(j))j is-an-element-of Zd the function SIGMA(j is-an-element-of Zd) y(j) B(./h - j). For people not used to this kind of approximation, this paper intends to do a summary of the main definitions, properties and utilizations of Schoenberg's approximation: we show that the main tool to handle Schoenberg's approximation is the Fourier transform of B and even more its modified version, the transfer function of B; we give conditions for convergence of SIGMA(j is-an-element-of Zd) f(jh)B(./h-j) when h tends to zero, and we give various ways to define various B as combinations of translates of some function phi (usually phi is either some radial function, or obtained by a tensor product of some radial function), depending on the properties we want for the associated Schoenberg's approximation. Last, we show how multi-resolution analysis, subdivision techniques, and wavelets techniques, are nicely connected to Schoenberg's approximation.