The wavelet transform is a tool that cuts up data or functions or operators into different frequency components, and then studies each component with a resolution matched to its scale. Forerunners of this technique were invented independently in pure mathematics (Calderón's resolution of the identity in harmonic analysis-see e.g., Calderón (1964)), physics (coherent states for the (ax + b)group in quantum mechanics, first constructed by Aslaksen and Klauder (1968), and linked to the hydrogen atom Hamiltonian by Paul (1985)) and engineering (QMF filters by Esteban and Galland (1977), and later QMF filters with exact reconstruction property by Smith and Barnwell (1986), Vetterli (1986) in electrical engineering; wavelets were proposed for the analysis of seismic data by J. Morlet (1983)). The last five years have seen a synthesis between all these different approaches, which has been very fertile for all the fields concerned.Let us stay for a moment within the signal analysis framework. (The discussion can easily be translated to other fields.) The wavelet transform of a signal evolving in time (e.g., the amplitude of the pressure on an eardrum, for acoustical applications) depends on two variables: scale (or frequency) and time; wavelets provide a tool for time-frequency localization. The first section tells us what time-frequency localization means and why it is of interest. The remaining sections describe different types of wavelets. Time-frequency localization.In many applications, given a signal f (t) (for the moment, we assume that t is a continuous variable), one is interested in its frequency content locally in time. This is similar to music notation, for example, which tells the player which notes (= frequency information) to play at any given moment. The standard Fourier transform,also gives a representation of the frequency content of f, but information concerning time-localization of, e.g., high frequency bursts cannot be read off easily from .F f . Time-localization can be achieved by first windowing the signal f, so Downloaded 10/05/12 to 132.206.27.25. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CHAPTER 1 as to cut off only a well-localized slice of f, and then taking its Fourier transform:This is the windowed Fourier transform, which is a standard technique for timefrequency localization. 1 It is even more familiar to signal analysts in its discrete version, where t and w are assigned regularly spaced values: t = nto , w = mwo , where m, n range over Z, and wo, to > 0 are fixed. Then (1.1.1) becomes Twin (f) = f ds ƒ(s) g(s -nto) e-imw08

We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted p -penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem. Use of such p -penalized problems with p < 2 is often advocated when one expects the underlying ideal noiseless solution to have a sparse expansion with respect to the basis under consideration. To compute the corresponding regularized solutions, we analyze an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm.

(1.1). Basic dilation factors different from 2 are possible: there exist orthonormal bases in which this factor is any rational p/q > 1 [5]; in more than one dimension we may even choose a dilation matrix instead of an isotropic dilation factor. In these more general cases, it may be necessary to introduce more than one (but always a finite number). We shall restrict ourselves to one dimension here, and to the dilation factor 2, as in (1.1). Bases with factor 2 are by far the easiest to implement for numerical computations.All interesting examples of orthonormal wavelet bases can be constructed via multiresolution analysis. This is a framework developed by Mallat [6] and Meyer [7], in which the wavelet coefficients (f, Ojk) for fixed j describe the difference between two approximations of f, one with resolution 2j-, and one with the coarser resolution 2 .The following succinct review of multiresolution analysis suffices for the understanding of this paper; for more details, examples, and proofs we refer the reader to [6] and [7].The successive approximation spaces V in a multiresolution analysis can be characterized by means of a scaling function ok. More precisely, we assume that the integer translates of b are an orthonormal basis for the space Vo, which we define to be the approximation space with resolution 1. The approximation spaces V with resolution 2 are then defined as the closed linear spans of the bk (k 7/), where (1.2) dpjk 2-J/adp(2-Jx-k).To ensure that projections on the V describe successive approximations, we require Vo c V_l, which implies (1.3)

Two different procedures are studied by which a frequency analysis of a time-dependent signal can be effected, locally in time. The first procedure is the short-time or windowed Fourier transform, the second is the "wavelet transform," in which high frequency components are studied with sharper time resolution than low frequency components. The similarities and the differences between these two methods are discussed. For both schemes a detailed study is made of the reconstruction method and its stability, as a function of the chosen time-frequency density. Finally the notion of "time-frequency localization" is made precise, within this framework, by two localization theorems.

Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise: to two dual Riesz bases of compactly supported wavelets. We give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions, and we present a sufficient condition for the decay of their Fourier transforms. We study the regularity of these biorthogonal bases. We provide several families of examples, all symmetric (corresponding to "linear phase" filters). In particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity; we also show how to construct symmetric biorthogonal wavelet bases "close" to a (nonsymmetric) orthonormal basis.