I. Daubechies, A. Grossman, and Y. Meyer. Painless nonorthogonal expansions. Journal Math. Phys., 27:1271--1283, November 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....completely. Then the question arises under which conditions on the sampling set a function f is uniquely and stably determined by the samples of W f and how to reconstruct f . The case of regular sampling is well understood. For band limited g regular sampling has been treated in detail in [4]. More generally, given an almost arbitrary function g, I. Daubechies has found nearly optimal estimates on the lattice density (ff; fi) so that f can be completely and stably reconstructed from W f(ffkfi ; fi ) j; k 2 ZZ, or from S f(ffj; fik) 3] Regular sampling is also treated in ....

....of wavelet and short time Fourier transforms we refer to [1, 6, 9] In this note we show how recent quantitative results on irregular sampling of band limited functions also yield irregular sampling theorems for wavelet and short time Fourier transforms. This extends the program of [3] and [4] to irregular sampling. The novelty lies (a) in the explicit constants and error estimates, b) in the introduction of adaptive weights to compensate for local variations of the sampling density, and (c) in an improved reconstruction algorithm. 2. Results: Suppose that the wavelet g is ....

[Article contains additional citation context not shown here]

I. Daubechies, A Grossman, and Y. Meyer. Painless nonorthogonal expansions. J. Math. Phys. 27 (1986), 1271-1283.

....our theory of GSI systems to one of its most important special cases, viz. wavelet systems. There were many contributions, during the last decade, to the study of the Bessel, frame and other related properties of wavelet systems. Examples of univariate wavelet frames could already be found in [DGM]; necessary and su#cient conditions for mother wavelets to generate frames were discussed (implicitly) in [Me] and [D2] Characterizations of univariate tight wavelet frames associated with integer dilation were established in [FGWW] and [HW] with the multivariate counterparts of these results ....

I. Daubechies, A. Grossmann, and Y. Meyer (1986), "Painless nonorthogonal expansions ", J. Mathematical Physics 27, 1271--1283.

....ubiquitous. They started as a mathematical theory by Duffin and Schaeffer [9] who provided an abstract framework for the idea of time frequency atomic decomposition by Gabor [10] The theory then laid largely dormant until 1986 with the publication of the work by Daubechies, Grossman and Meyer [11]. Since then, frames have evolved into a state of the art signal processing tool. The mathematics of frames can be found in several excellent sources. The original work by Duffin and Schaeffer introduced frames for Hilbert spaces [9] The paper by Daubechies, Grossman and Meyer [11] discusses ....

....and Meyer [11] Since then, frames have evolved into a state of the art signal processing tool. The mathematics of frames can be found in several excellent sources. The original work by Duffin and Schaeffer introduced frames for Hilbert spaces [9] The paper by Daubechies, Grossman and Meyer [11] discusses applications to wavelet and Gabor transforms. A beautiful tutorial on the art of frame theory was written by Casazza [12] Some particular classes of frames have been extensively studied: Gabor frames (also called WeylHeisenberg frames) are described by Heil and Walnut in [13] and by ....

I. Daubechies, A. Grossman, and Y. Meyer. Painless nonorthogonal expansions. Journ. Math. Phys., 27:1271--1283, November 1986.

....of interest, and, in fact, our initial development of the frame material in [RS1 3] was done from scratch . While this somewhat cavalier approach might have had its own advantages, it also, inevitably, resulted in the re invention of known and even classical results (the Zak transform and the [DGM] painless construction of WH frames were among our early innovations ) Communications we had in late 1992 with Chris Heil had helped us in drawing connections between our work and the rich frame literature. Our first presentation of the duality principle (in Oberwolfach, Summer 1993) had led to ....

....well known, corollaries are derived below as an illustration of the e#cacy of the duality principle. Further corollaries and applications of the duality principle are collected in 3.1. These other ones include (a) review and extensions of the painless construction of WH tight frames from [DGM], b) results on frames generated by functions that are non negative or that their Fourier transform is such, c) partial orthogonality relations that a tight WH frame of a special structure, dubbed sup adjoint herein, must satisfy, and more. Corollary 2.5. A self adjoint X is a fundamental ....

[Article contains additional citation context not shown here]

I. Daubechies, A. Grossmann, and Y. Meyer, Painless non-orthogonal expansions, J. Math. Phys., 45 (1986), 1271-1283.

....such that wj,k spans L2(IR) Clearly such a frequency decomposition is very similar to the STFT, in that the channels are of uniform effective bandwidth and hence useful. Later we will see how to construct many such frequency decompositions. Even though orthonormal DSTFT bases can be constructed [8, 32, 10], there is no general technique to obtain such decompositions with windows that have both desirable localization properties and good numerical algorithms associated with them. Usually one starts with a desirable w(t) wj,k not necessarily orthonormal) 5, 15] and hopes that expansions like Eqn. 7 ....

I. Daubechies, A. Grossman, and Y. Meyer. Painless non-orthogonal expansions. dourml of Mathematical Phi/sics, 27(5):1271 1283, 1986.

....the fg [mN;n] g corresponding to frame parameters (1; N) is a proper subset of fg [m;n] g, it follows that fg [mN;n] g cannot be a frame either. For frame parameters (a; b) with 0 ab 1 it is not difficult to construct W H frames fg [ma;nb] g such that g 2 L 2 (R) is a smooth function [DGM] H3] HW] Taking the case a = 1; b = 1 2 , for example, let v be an infinitely differentiable function on R such that v(x) ae 0 if x 0 1 if x 1 82 WILLARD MILLER JR. and 0 v(x) 1 if 0 x 1. Set (7.23) g(x) 8 : 0; x 0 v(x) 0 x 1 [1 Gamma v 2 (x ....

....8.2 Affine frames. The general definitions and analysis of frames presented in Chapter 7 clearly apply to wavelets. However, there is no affine analog of the WeilBrezin Zak transform which was so useful for Weyl Heisenberg frames. Nonetheless we can prove the following result directly. Lemma 8. 1 [DGM] Let g 2 L 2 (R) such that the support of Fg is contained in the interval [ L] where 0 L 1, and let a 0 1; b 0 0 with (L Gamma )b 0 1. Suppose also that 0 A X m jFg(a m 0 y)j 2 B 1 84 WILLARD MILLER JR. for almost all y 0. Then fg mn g is a frame for H ....

]I. Daubechies, A. Grossman, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271--1283.

.... fx n g suchthat 8f 2H# f = X n hf#x n ix n = X n hf#x n ix n : All duals fx n g havebeencharacterized and formulated in an algebraic and parametric formula, e.g. 16] 18] References on frames goes back to Duffin and Shaeffer s original work [11] Others includes [1] 9] [10], 12] 14] 25] In general, a frame is overcomplete.Frames have the redundancy property, natural to application problems where robustness, error tolerance and noise suppression play a vital role, e.g. 3] 9] 20] 21] Frame conditions can be further expnaded. Example 2.1 of Section 2 is ....

I. Daubechies, A. Grossmann, and Y. Meyer. Painless nonorthogonal expansions. J. Math. Phys., 27: pp. 1271--1283, 1986.

.... and that of Feichtinger (joined later on by Grochenig) focusing on the more functional analytic (modulation spaces) and group theoretic aspects of Gabor expansions [12] 13] A major development in the mathematical theory of Gabor expansions was made in 1986 by Daubechies, Grossmann and Meyer [14] who placed the Gabor expansion problem in the context of frames for a Hilbert space. The latter concept was introduced by Duffin and Schaeffer [15] in 1951 for addressing completeness and expansion problems involving sets of exponentials in spaces of band limited functions. For a Gabor system (g; ....

I. Daubechies, A. Grossmann and Y. Meyer, Painless non-orthogonal expansions, J. Math. Phys., 27 (1986), pp. 1271--1283.

....ubiquitous. They started as a mathematical theory by Duffin and Schaeffer [13] who provided an abstract framework for the idea of time frequency atomic decomposition by Gabor [15] The theory then laid largely dormant until 1986 with the publication of work by Daubechies, Grossman and Meyer [11]. Since then, frames have evolved into a state of the art signal processing tool. The mathematics of frames can be found in several excellent sources. The original work by Duffin and Schaeffer introduced frames for Hilbert spaces [13] The paper by Daubechies, Grossman and Meyer [11] discusses ....

....and Meyer [11] Since then, frames have evolved into a state of the art signal processing tool. The mathematics of frames can be found in several excellent sources. The original work by Duffin and Schaeffer introduced frames for Hilbert spaces [13] The paper by Daubechies, Grossman and Meyer [11] discusses applications to wavelet and Gabor transforms. A beautiful tutorial on the art of frame theory was written by Casazza [4] Some particular classes of frames have been extensively studied: Gabor (also called Weyl Heisenberg) frames are described by Heil and Walnut in [22] while the paper ....

I. Daubechies, A. Grossman, and Y. Meyer. Painless nonorthogonal expansions. Journ. Math. Phys., 27:1271--1283, 1986.

....1985, Y. Meyer, an harmonic analyst, pointed out the strong connection with the existing analysis techniques of singular integral operators. Ingrid Daubechies became involved in 1986 and this started an interaction between signal analysis and the mathematical aspects of dilations and translations [7, 10]. Also Stephane Mallat became involved when he noticed the connection with multiresolution analysis [22] A major breakthrough was provided in 1988 when Daubechies managed to construct a family of orthonormal wavelets with compact support [8] a result inspired by the work of Meyer and Mallat in ....

I. Daubechies, A. Grossmann, and Y. Meyer. Painless nonorthogonal expansions. J. Math. Phys., 27(5):1271--1283, 1986. 44 A. Bultheel

....and the Theory of Frames. We shall brie y describe what is new here that which is not merely an automatic consequence of existing ideas. First, there is of course a general machinery for getting continuous reproducing formulas like (1) via the theory of square integrable group representations [11, 8]. Such a theory has been applied to develop wavelet like representations over groups other than the usual ax b group on R n , see [3] However, the particular geometry of ridge functions does not allow the identi cation of the action of on with a linear group representation (notice that the ....

....requires only one vanishing moment in any dimension. Second, in constructing frames of ridgelets, we have been guided by the theory of wavelets, which holds that one can turn continuous transforms into discrete expansions by adopting a strategy of discretizing frequency space into dyadic coronae [7, 8]; this goes back to Littlewood5 Paley [13] Our approach indeed uses such a strategy for dealing with the location and scale variables in the d dictionary. However, in dealing with ridgelets there is also an issue of discretizing the directional variable u that seems to be a new element: u must ....

I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271-1283.

....the Theory of Frames. We shall briefly describe what is new here that which is not merely an automatic consequence of existing ideas. First, there is of course a general machinery for getting continuous reproducing formulas like (1) via the theory of square integrable group representations [11, 8]. Such a theory has been applied to develop wavelet like representations over groups other than the usual ax b group on R n , see [3] However, the particular geometry of ridge functions does not allow the identification of the action of Gamma on with a linear group representation (notice ....

....requires only one vanishing moment in any dimension. Second, in constructing frames of ridgelets, we have been guided by the theory of wavelets, which holds that one can turn continuous transforms into discrete expansions by adopting a strategy of discretizing frequency space into dyadic coronae [7, 8]; this goes back to Littlewood5 Paley [13] Our approach indeed uses such a strategy for dealing with the location and scale variables in the Gamma d dictionary. However, in dealing with ridgelets there is also an issue of discretizing the directional variable u that seems to be a new element: ....

I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271--1283.

....The notions of mother framelets, tight framelets, etc. have then their obvious meaning. Some historical pointers: The concept of frames was first introduced by Du#n and Schae#er in [DS] Examples of univariate wavelet frames can already be found in the work of Daubechies, Grossmann and Meyer, [DGM]; necessary and su#cient conditions for mother wavelets to generate frames are implicit in e.g. Me] and [D2] Characterizations of univariate tight wavelet frames are implicit in the work of Wang and Weiss, FGWW, HW] An explicit characterization of tight wavelet frames (in the multivariate case) ....

I. Daubechies, A. Grossmann, and Y. Meyer (1986), "Painless nonorthogonal expansions ", J. Mathematical Physics 27, 1271--1283.

.... re normalization of the g # X(#) one then has # g#X(#) #f, g# 2 = #f# 2 L2 (IR d ) for all f # L 2 (IR d ) This tight frame condition is equivalent to the perfect reconstruction property f = # g#X(#) #f, g# g, for all f # L 2 (IR d ) We refer to [BL] CS] D2] [DGM], DS] and [RS1 5] for more information about frames in general and wavelet frames in particular. We are interested in the study of wavelet frames that are derived from a multiresolution analysis (MRA) Although our results and observations cover the case of vector (FSI) MRA, we restrict our ....

.... MRA In particular, can one construct framelets from the MRA induced by a univariate B spline or a multivariate box spline # A few remarks about the literature concerning wavelet frames: examples of univariate wavelet frames can already be found in the work of Daubechies, Grossmann and Meyer, [DGM]; necessary and su#cient conditions for mother wavelets to generate frames are implicit in e.g. Me] and [D2] Characterizations of univariate tight wavelet frames are implicit in the work of X. Wang and G. Weiss, FGWW] HW] who actually characterized univariate orthonormal wavelet bases. An ....

I. Daubechies, A. Grossmann, and Y. Meyer (1986), "Painless nonorthogonal expansions ", J. Mathematical Physics 27, 1271--1283.

....and go back to the context of subband filters, or more precisely quadrature mirror filters [36, 37, 41, 51, 52, 53, 54, 58, 56, 59] In mathematical analysis, wavelets were defined as translates and dilates of one fixed function and were used to both analyze and represent general functions. [15, 20, 25, 35, 24]. In the late eighties the introduction of multiresolution analysis and the fast wavelet transform by Mallat and Meyer provided the connection between subband filters and wavelets [33, 34, 35] this led to the first construction of smooth, orthogonal, and compactly supported wavelets in 1987 [18] ....

I. Daubechies, A. Grossmann, and Y. Meyer. Painless nonorthogonal expansions. J. Math. Phys., 27(5):1271-- 1283, 1986.

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I. Daubechies, A. Grossman, and Y. Meyer. Painless nonorthogonal expansions. Journal Math. Phys., 27:1271--1283, November 1986.

No context found.

Daubechies, I., Grossmann, A., and Meyer, Y. (1986). Painless nonorthogonal expansions. J. Math. Phys. 27, 1271--1283.

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I. Daubechies, A. Grossmann, and Y. Meyer. Painless nonorthogonal expansions. J. Math. Phys., 27(5):1271-1283, 1986.

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I. Daubechies, A Grossman, and Y. Meyer. Painless nonorthogonal expansions. J. Math. Phys. 27 (1986), 1271-1283.

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I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271--1283.

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Daubechies, I., A. Grossmann, and Y. Meyer. Painless nonorthogonal expansions. J. Math. Phys. 27 (1986), 1271-1283.

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I. Daubechies, A. Grossmann, and Y. Meyer. Painless nonorthogonal expansions. J. Math. Phys., 27(5):1271--1283, 1986.

No context found.

I. Daubechies, A. Grossman, and Y. Meyer. Painless nonorthogonal expansions. Journal Math. Phys., 27:1271--1283, November 1986.

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Daubechies, I., Grossmann, A., and Meyer, Y. (1986). Painless nonorthogonal expansions. J. Math. Phys. 27, 1271--1283.

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I. Daubechies, A. Grossmann, and Y. Meyer, \Painless nonorthogonal expansions," Journal of Mathematical Physics 27, pp. 1271-1283, 1986.

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