Using the refinement equation for the construction of pre-wavelets

Abstract

A variety of methods have been proposed for the construction of wavelets. Among others, notable contributions have been made by Battle, Daubechies, Lemarié, Mallat, Meyer, and Stromberg. This effort has led to the attractive mathematical setting of multiresolution analysis as the most appropriate framework for wavelet construction. The full power of multiresolution analysis led Daubechies to the construction of compactly supported orthonormal wavelets with arbitrarily high smoothness. On the other hand, at first sight, it seems some of the other proposed methods are tied to special constructions using cardinal spline functions of Schoenberg. Specifically, we mention that Battle raises some doubt that his block spin method "can produce only the Lemarié Ondelettes". A major point of this paper is to extend the idea of Battle to the generality of multiresolution analysis setup and address the easier job of constructing pre-wavelets from multiresolution. © 1991 J.C. Baltzer A.G. Scientific Publishing Company.