Affine frame decompositions and shift-invariant spaces

In this paper, we show that the property of tight affine frame decomposition of functions in L-2 can be extended in a stable way to functions in Sobolev spaces when the generators of the tight affine frames satisfy certain mild regularity and vanishing moment conditions. Applying the affine frame operators Q(j) on jth levels to any function f in a Sobolev space reveals the detailed information Q(j) f of f in such tight affine decompositions. We also study certain basic properties of the range of the affine frame operators Q(j) such as the topological property of closedness and the notion of angles between the ranges for different levels, and thus establishing some interesting connection to (tight) frames of shift-invariant spaces. (C) 2005 Elsevier Inc. All rights reserved.