. The Balian--Low theorem (BLT) is a key result in time-frequency analysis, originally stated by Balian and, independently, by Low, as: If a Gabor system fe 2ßimbt g(t \Gamma na)g m;n2Z with ab = 1 forms an orthonormal basis for L 2 (R), then `Z 1 \Gamma1 jt g(t)j 2 dt ' `Z 1 \Gamma1 jfl g(fl)j 2 dfl ' = +1: The BLT was later extended from orthonormal bases to exact frames. This paper presents a tutorial on Gabor systems, the BLT, and related topics, such as the Zak transform and Wilson bases. Because of the fact that (g 0 ) (fl) = 2ßifl g(fl), the role of differentiation in the proof of the BLT is examined carefully. The major new contributions of this paper are the construction of a complete Gabor system of the form fe 2ßibm t g(t \Gamma an )g such that f(an ; bm )g has density strictly less than 1, an Amalgam BLT that provides distinct restrictions on Gabor systems fe 2ßimbt g(t \Gamma na)g that form exact frames, and a new proof of the BLT for exact frame...

Discrete affine systems are obtained by applying dilations to a given shift-invariant system. The complicated structure of the affine system is due, first and foremost, to the fact that it is not invariant under shifts. Affine frames carry the additional difficulty that they are "global" in nature: it is the entire interaction between the various dilation levels that determines whether the system is a frame, and not the behaviour of the system within one dilation level. We completely unravel the structure of the affine system with the aid of two new notions: the affine product, and a quasi-affine system. This leads to a characterization of affine frames; the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the Fourier domain. Several results, such as a general oversampling theorem follow from these characterizations. Most importantly, the affine product can be factored during a multiresolution analysis con...

: In this note we show that there exist wavelet frames that have nice dual wavelet frames, but for which the canonical dual frame does not consist of wavelets, i.e., can not be generated by the translates and dilates of a single function. Key words: Wavelet frame, the canonical dual frame, dual wavelet frame. 2000 AMS subject classication: 42C40, 42C15, 41A15. 2 1 Introduction We start by recalling some notations and denitions. Let H be a Hilbert space. A set of elements fh k g k2Z in H is said to be a frame (see [5]) in H if there exist two positive constants A and B such that Akfk 2 X k2Z jhf; h k ij 2 Bkfk 2 8 f 2 H; (1.1) where A and B are called the lower frame bound and upper frame bound, respectively. In particular, when A = B = 1, we say that fh k g k2Z is a (normalized) tight frame in H. The frame operator S : H ! H, which is associated with a frame fh k g k2Z , is dened to be Sf := X k2Z hf; h k ih k ; f 2 H: (1.2) It is evident that fh k g k2Z is...

Abstract. Continuous wavelets are often studied in the general framework of representation theory of square-integrable representations, or by using convolution relations and Fourier transforms. We consider the well-known problem whether these continuous wavelets can be discretized to yield wavelet frames. In this paper we us Calderón-Zygmund singular integral operators and atomic decompositions on spaces of homogeneous type, endowed with families of general translations and dilations, to attack this problem, and obtain strong convergence results for wavelets expansions in a variety of classical functional spaces and smooth molecule spaces. This approach is powerful enough to yield, in a uniform way, for example, frames of smooth wavelets for matrix dilations in R n, for an affine extension of the Heisenberg group, and on many commutative hypergroups. 1.

. 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 first part, the fiberization of the analysis operator of a shift-invariant system is discussed. That fiberization 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 refinable 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 refinable functions, including their smoothness and the convergence of the cascade and subdivision algorithms. In particular, it follows that these latter a...

. For groups that are semidirect products of a vector group and a unimodular group, we prove that the existence of a discrete frame associated with a unitary representation implies the square-integrability of the representation. 1. Introduction In this paper we consider a unimodular locally compact second countable (lcsc) topological group H and a continuous representation # of H acting in R n . We define G as the semidirect product of R n and H with respect to the action given by #, i.e. G = R n # H and (x, h)(x # , h # ) = (x + # h x # , hh # ), where (x, h) # G and (x # , h # ) # G. Moreover, we consider a unitary continuous representation U of G acting in a complex separable Hilbert space H with scalar product #, #, linear in the first argument. With slight abuse of notation, the representation U is said to be square-integrable if there is # # H, # #= 0 such that, for all # # H # G | ##, U g ## | 2 dG (g) < #, and such a # is said to be an admissible ...

. Every m th order cardinal spline wavelet is a linear combination of the functions fN (l) m+l (2x \Gamma j); j 2 Zg. Here the function Nm is the m th order cardinal B-spline. In this paper we prove that the single function N (l) m+l (2x), or N (l) m+l (2x \Gamma 1) is a wavelet when m and l satisfy some mild conditions. As l decreases, so does the support of the wavelet. When l increases, the smoothness of the dual wavelet improves. Each wavelet is constructed by spline multiresolution analysis. The dual multiresolution analyses are given. 1. Introduction The simplest example of an orthonormal spline wavelet basis is the Haar basis. The orthonormal spline wavelet bases of higher order spline wavelets were given by Battle [2] and Lemari'e [20] by using different methods. Cohen, Daubechies, and Feauveau constructed biorthogonal wavelet bases of compactly supported wavelets [11,13]. The most important advantage of Cohen, Daubechies, and Feauveau's construction is that both wa...

In principle it is well known that for sufficiently nice wavelet functions the regularity of the wavelet transform allows to recover any L²-function from its samples over any sufficiently dense, irregular sampling set. Equivalently, the (irregular) set of affine transforms of the given wavelet function forms a frame for L²(R d). In the present paper a systematic treatment of sufficient conditions for the validity of such statement is provided, on the basis of two new Banach spaces of functions, denoted by F0 and F1 in the sequel, using classical concepts (e.g. norms involving derivatives etc.). The norms on these spaces also turn out to be highly suitable for the description of perturbation results. Given an irregular wavelet frame using an atom from one of these spaces implies that for any sufficiently close irregular set (in the sense of small jitter error), and sufficiently small modification of the atom (in terms of one of the two norms). Whereas it is shown that the perturbation may occur in the sense that every parameter is allowed to be perturbed in the same size for

The first wavelet system was discovered by Alfréd Haar one hundred years ago. Since then the field has grown enormously. In 1952, Richard Duffin and Albert Schaeffer synthesized the earlier ideas of a number of illustrious mathematicians into a unified theory, the theory of frames. Interest in frames as intriguing objects in their own right arose when wavelet theory began to surge in popularity. Wavelet and frame analysis is found in such diverse fields as data compression, pseudo-differential operator theory and applied statistics. We shall explore five areas of frame and wavelet theory: frame bound gaps, smooth Parseval wavelet frames, generalized shearlets, Grassmannian fusion frames, and p-adic wavlets. The phenomenon of a frame bound gap occurs when certain sequences of functions, converging in L2 to a Parseval frame wavelet, generate systems with frame bounds that are uniformly bounded away from 1. In the 90’s, Bin Han proved the existence of Parseval wavelet frames which are smooth and compactlysupported on the frequency domain and also approximate wavelet set wavelets. We discuss problems that arise when one attempts to generalize his results to higher dimensions. A shearlet