Etude de la vitesse de convergence de l'algorithme en cascade dans la construction des ondelettes d'Ingrid Daubechies

Abstract

The aim of this paper is the study of the convergence of algorithms involved in the resolution of two scale equations. They are fixed point algorithms, often called "cascade algorithm", which are used in the construction of wavelets. We study their speed of convergence in Lebesgue and Besov spaces, and show that the quality of the convergence depends on two independent factors. The first one, as we could foresee, is the regularity of the scaling function which is the solution of the equation. The second factor (that is the essential discovery of this work) concerns specific algebraic properties of the function used to initialize the algorithm. This function must satisfy conditions analogous to Strang-Fix conditions.