This paper is an expository survey of results on integral representations and discrete sum expansions of functions in L 2 (R) in terms of coherent states. Two types of coherent states are considered: Weyl–Heisenberg coherent states, which arise from translations and modulations of a single function, and affine coherent states, called “wavelets,” which arise as translations and dilations of a single function. In each case it is shown how to represent any function in L²R) as a sum or integral of these states. Most of the paper is a survey of literature, most notably the work of I. Daubechies, A. Grossmann, and J. Morlet. A few results of the authors are included.

this paper is somewhat dierent, in that we are concerned with the connection between density properties of and frame properties of S(g; ), and the analogous problem for systems T (g; ) of pure translates. For the case of Gabor systems, there is a rich literature on this subject, especially when is the rectangular lattice = aZ

Banach frames and atomic decompositions are sequences that have basis-like properties but which need not be bases. In particular, they allow elements of a Banach space to be written as linear combinations of the frame or atomic decomposition elements in a stable manner. In this paper we prove several functional-analytic properties of these decompositions, and show how these properties apply to Gabor and wavelet systems. We first prove that frames and atomic decompositions are stable under small perturbations. This is inspired by corresponding classical perturbation results for bases, including the Paley-Wiener basis stability criteria and the perturbation theorem of Kato. We introduce new and weaker conditions which ensure the desired stability. We then prove duality properties of atomic decompositions and consider some consequences for Hilbert frames. Finally, we demonstrate how our results apply in the practical case of Gabor systems in weighted L² spaces. Such systems can form...

We investigate the properties an exotic symbol class of pseudodifferential operators, Sjöstrand's class, with methods of time-frequency analysis (phase space analysis). Compared to the classical treatment, the time-frequency approach leads to striklingly simple proofs of Sjöstrand's fundamental results and to far-reaching generalizations.

Abstract. This paper is a survey of research in discrete expansions over the last 10 years, mainly of functions in L 2 (R). The concept of an orthonormal basis {fn}, allowing every function f ∈ L 2 (R) to be written f = ∑ cnfn for suitable coefficients {cn}, is well understood. In separable Hilbert spaces, a generalization known as frames exists, which still allows such a representation. However, the coefficients {cn} are not necessarily unique. We discuss the relationship between frames and Riesz bases, a subject where several new results have been proved over the last 10 years. Another central topic is the study of frames with additional structure, most important Gabor frames (consisting of modulated and translated versions of a single function) and wavelets (translated and dilated versions of one function). Along the way, we discuss some possible directions for future research. 1.

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Abstract. The refinement equation ϕ(t) = �N2 k=N1 ckϕ(2t − k) playsakey role in wavelet theory and in subdivision schemes in approximation theory. Viewed as an expression of linear dependence among the time-scale translates |a | 1/2ϕ(at − b)ofϕ∈L2 (R), it is natural to ask if there exist similar dependencies among the time-frequency translates e2πibtf(t + a) off∈L2 (R). In other words, what is the effect of replacing the group representation of L2 (R) induced by the affine group with the corresponding representation induced by the Heisenberg group? This paper proves that there are no nonzero solutions to lattice-type generalizations of the refinement equation to the Heisenberg group. Moreover, it is proved that for each arbitrary finite collection {(ak,bk)} N k=1, the set of all functions f ∈ L2 (R) such that {e2πibktf(t+ ak)} N k=1 is independent is an open, dense subset of L2 (R). It is conjectured that this set is all of L2 (R) \{0}. 1.

This work develops a quantitative framework for describing the overcompleteness of a large class of frames. A previous article introduced notions of localization and approximation between two frames F = {fi}i∈I and E = {ej}j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of E via a map a: I → G. This article shows that those abstract results yield an array of new implications for irregular Gabor frames. Additionally, various Nyquist density results for Gabor frames are recovered as special cases, and in the process both their meaning and implications are clarified. New results are obtained on the excess and overcompleteness of Gabor frames, on the relationship between frame bounds and density, and on the structure of the dual frame of an irregular Gabor frame. More generally, these results apply both to Gabor frames and to systems of Gabor molecules, whose elements share only a common envelope of concentration in the time-frequency plane. The notions of localization and related approximation properties are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. In this article, a comprehensive examination of the interrelations among these localization and approximation concepts is made, with most implications shown to be sharp.

Abstract — We consider three different versions of the Zak transform (ZT) for discrete-time signals, namely, the discretetime ZT, the polyphase transform, and a cyclic discrete ZT. In particular, we show that the extension of the discrete-time ZT to the complex �-plane results in the polyphase transform, an important and well-known concept in multirate signal processing and filter bank theory. We discuss fundamental properties, relations, and transform pairs of the three discrete ZT versions, and we summarize applications of these transforms. In particular, the discrete-time ZT and the cyclic discrete ZT are important for discrete-time Gabor expansion (Weyl–Heisenberg frame) theory since they diagonalize the Weyl–Heisenberg frame operator for critical sampling and integer oversampling. The polyphase representation plays a fundamental role in the theory of filter banks, especially

Abstract. We provide a detailed development of a function valued inner product known as the bracket product and used effectively by de Boor, Devore, Ron and Shen to study translation invariant systems. We develop a version of the bracket product specifically geared to Weyl-Heisenberg frames. This bracket product has all the properties of a standard inner product including Bessel’s inequality, a Riesz Representation Theorem, and a Gram-Schmidt process which turns a sequence of functions (gn) into a sequence (en) with the property that (Emben)m,n∈Z is orthonormal in L2 (R). Armed with this inner product, we obtain several results concerning Weyl-Heisenberg frames. First we see that fiberization in this setting takes on a particularly simple form and we use it to obtain a compressed representation of the frame operator. Next, we write down explicitly all those functions g ∈ L2 (R) and ab = 1 so that the family (EmbTnag) is complete in L2 (R). One consequence of this is that for functions g supported on a half-line [α, ∞) (in particular, for compactly supported g), (g, 1, 1) is complete if and only if sup0≤t<a|g(t − n) | ̸ = 0 a.e. Finally, we give a direct proof of a result hidden in the literature by proving: For any g ∈ L2 (R), A ≤ ∑ n |g(t − na)|2 ≤ B is equivalent to (Em/ag) being a Riesz basic sequence. 1.