Multiscale Characterizations of Besov Spaces on Bounded Domains

The theory of wavelets and multiresolution analysis is usually developed on R. However, applications of wavelets to image processing and numerical methods for partial differential equations require multiresolution analysis on domains or manifolds in R. The study of multiresolution in these settings is just beginning. Building on the construction of multiresolution on intervals (and cubes in R ), Cohen et al. [2] have constructed a multiresolution which applies to a fairly large class of domains 0 (basically coordinatewise Lipschitz) in R. They have shown in their analysis that various smoothness spaces can be characterized by this multiresolution. For example, their analysis applies to the Besov spaces Bq((Lp)(0) provided p 1. However, the same Besov spaces with p<1 are also important in analysis, especially in analyzing nonlinear methods [5] such as image compression [4] or noise removal [10]. The purpose of the present paper is to show that the Besov spaces for p<1 can also be characterized in the usual way by multiresolution analysis. To prove this, we analyze the approximation properties of certain sequences of operators Tj , j # N, which include as a special case the projectors in the Cohen et al. multiresolution. The operators take the form