Deslauriers-dubuc: Ten Years After

Ten years ago, Deslauriers and Dubuc introduced a process for interpolating data observed at the integers, producing a smooth function deened on the real line. In this note we point out that their idea admits many fruitful generalizations including: Interpolation of other linear functionals of f (not just point values), yielding other reenement schemes and biorthogonal wavelet transforms; Interpolation of vector-valued data, yielding vector reenement schemes and multi-wavelets; Interpolation of nonlinear functionals, yielding nonlinear reenement schemes and non-linear wavelet transforms. Interpolation by other families than polynomials, yielding e.g. segmented interpolation and segmented transforms. We refer to a variety of recent work on these schemes. Suppose we are given the point-values (k = f(k) : k 2 Z) of a function f evaluated at the integers. We wish to interpolate, getting a function ~ f(x) deened for x 2 R, obeying ~ f(k) = k. introduced a multiscale reenement technique for this problem. For integer L 0, let D = 2L + 1 be an odd integer > 0. They obtained such a function ~ f by interpolating the data at the integers to a function deened on the binary rationals by repeated application of a two-scale reenement transformation. If ~ f has already been deened at all binary rationals with denominator 2 j , j 0, their process extends it to all binary rationals with denominator 2 j+1 , i.e. all points halfway between previously deened points. Speciically, to deene the function at (k + 1=2)=2 j when it is already deened at all k=2 j , we t a polynomial j;k to the data (k 0 =2 j ; ~ f(k 0 =2 j)) for k 0 2 f(k ? L)=2 j ; : : : ; (k + L + 1)=2 j g { this polynomial is unique { and set ~ f((k + 1=2)=2 j) j;k ((k + 1=2)=2 j): For D = 1 this is just linear interpolation, pure and simple. Dubuc showed that this scheme deenes a function which is uniformly continuous at the rationals and hence has a unique continuous extension to the reals. They showed that this extension is actually C R for R = R(D) an increasing function of D = 2L + 1. We illustrate the basic calculation of the local polynomial j;k in Figure 1 using L = 1 and so D = 3. We illustrate the sequence of reenements …