Local Decomposition of Refinable Spaces and Wavelets

Abstract

A convenient setting for studyingmultiscale techniquesin various areas of applications is usually based on a sequence of nested closed subspaces of some function space script F which is often referred to asmultiresolution analysis.The concept ofwaveletsis a prominent example where the practical use of such a multiresolution analysis relies on explicit representations of theorthogonaldifference between any two subsequent spaces. However, many applications prohibit the employment of a multiresolution analysis based on translation invariant spaces on all of open face Rs, say. It is then usually difficult to compute orthogonal complements explicitly. Moreover, certain applications suggest using other types of complements, in particular, those corresponding tobiorthogonal wavelets.The main objective of this paper is therefore to study possibly nonorthogonal but in a certain sensestableand evenlocaldecompositions of nested spaces and to develop tools which are not necessarily confined to the translation invariant setting. A convenient way ofparametrizingsuch decompositions is to reformulate them in terms of matrix relations. This allows one to characterizeallstable or local decompositions by identifying unique matrix transformations that carry one given decomposition into another one. It will be indicated how such a mechanism may help realizing several desirable features of multiscale decompositions and constructing stable multiscale bases with favorable properties. In particular, we apply these results to the identification of decompositions induced by local linear projectors. The importance of this particular application with regard to the construction of multiscale Riesz bases will be pointed out. Furthermore, we indicate possible specializations to orthogonal decompositions of spline spaces relative to nonuniform knot sequences, piecewise linear finite elements and principal shift invariant spaces. The common ground of all these examples as well as other situations of practical and theoretical interest is thatsome initialmultiscale basis is available. The techniques developed here can then be used to generate from such an initial decomposition other ones with desirable properties pertaining to moment conditions or stability properties.