Local Decomposition of Reenable Spaces and Wavelets

A convenient setting for studying multiscale techniques in various areas of applications is usually based on a sequence of nested closed subspaces of some function space F which is often referred to as multiresolution analysis. The concept of wavelets is a prominent example where the practical use of such a mul-tiresolution analysis relies on explicit representations of the orthogonal diierence between any two subsequent spaces. However, many applications prohibit the employment of a multiresolution analysis based on translation invariant spaces on all of I R s , say. It is then usually diicult to compute orthogonal complements explicitly. Moreover, certain applications suggest using other types of complements , in particular, those corresponding to biorthogonal wavelets. The main objective of this paper is therefore to study possibly non-orthogonal but in a certain sense stable and even local decompositions of nested spaces and to develop tools which are not necessarily connned to the translation invariant setting. A convenient way of parametrizing such decompositions is to reformulate them in terms of matrix relations. This allows one to characterize all stable or local de-compositions 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 identiication 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 non-uniform knot sequences, piecewise linear nite elements and principal shift invariant spaces. The common ground of all these examples as well as other situations of practical and theoretical interest is that some initial multiscale 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.