Wavelets and Their Associated Operators

This article is devoted to the study of wavelets based on the theory of shift-invariant spaces. It consists of two, essentially disjoint, parts. In the rst part, the berization of the analysis operator of a shift-invariant system is discussed. That berization applies to wavelet systems via the notion of quasi-wavelet systems, and leads to the theory of wavelet frames. Highlights in this theory are the unitary and mixed extension principles, and the MRA construction of framelets. The second part of the article is devoted to the study of the cascade/transfer operators and the subdivision operator associated with a re nable function. The analysis there is primarily based on the interpretation of the cascade operator as a special quasi-interpolation scheme. This leads to a surprisingly simple analysis of certain properties of re nable functions, including their smoothness and the convergence of the cascade and subdivision algorithms. In particular, it follows that these latter algorithms, if handled properly, always converge. 1. Preface: Wavelets and Their Associated Operators This article advocates the analysis of wavelet systems via the study of their associated operators. The goal is neither to survey the current state-of-the-art in this area, nor to provide the reader with in-depth comprehensive analysis of any of the issues addressed. Rather, my attempt is to provide a glimpse into various contemporary aspects of wavelets, in a way that may whet the reader's appetite for further reading. Based on this philosophy, I have chosen setups that simplify the discussion even in cases when the simpli cation is purely notational. The notion of `the operators associated with a wavelet system' is so broad that it allows me to discuss two essentially disjoint topics. The rst topic concerns the intrinsic operators of the wavelet system: analysis and synthesis, with the main aim being to review the recent developments in the area of wavelet frames (cf. [66], [67], [68], [69], [70], [71], [72], [34], [22], [30]). The second topic is the analysis of the corresponding re nable/scaling function(s), a topic that is also pertinent to the area of uniform subdivision algorithms (cf. [56], [33], [15], [32]). The relevant operators in this discussion are the Approximation Theory IX 1 Charles K. Chui and Larry L. Schumaker (eds.), pp. 1{35. Copyright oc 1998 by Vanderbilt University Press, Nashville, TN.