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60
Data compression and harmonic analysis
 IEEE Trans. Inform. Theory
, 1998
"... In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon’s R(D) theory... ..."
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Cited by 140 (24 self)
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In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon’s R(D) theory...
Modulation Spaces on Locally Compact Abelian Groups
 University of Vienna
, 1983
"... this paper, which is now better accessible, in the same way as to the original report.) Hans G. Feichtinger 1 ..."
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Cited by 58 (2 self)
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this paper, which is now better accessible, in the same way as to the original report.) Hans G. Feichtinger 1
Resolution of the Wavefront Set using Continuous Shearlets
, 2008
"... Abstract. It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f. However, the Continuous Wavelet Transform is unabl ..."
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Cited by 37 (20 self)
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Abstract. It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f. However, the Continuous Wavelet Transform is unable to describe the geometry of the set of singularities of f and, in particular, identify the wavefront set of a distribution. In this paper, we employ the same framework of affine systems which is at the core of the construction of the wavelet transform to introduce the Continuous Shearlet Transform. This is defined by SHψf(a, s, t) = 〈f, ψast〉, where the analyzing elements ψast are dilated and translated copies of a single generating function ψ. The dilation matrices form a twoparameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements {ψast} form a system of smooth functions at continuous scales a> 0, locations t ∈ R 2, and oriented along lines of slope s ∈ R in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution f. 1.
NDimensional Affine WeylHeisenberg Wavelets
, 1993
"... : ndimensional coherent states systems generated by translations, modulations, rotations and dilations are described. Starting from unitary irreducible representations of the ndimen sional affine WeylHeisenberg group, which are not squareintegrable, one is led to consider systems of coherent s ..."
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Cited by 14 (1 self)
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: ndimensional coherent states systems generated by translations, modulations, rotations and dilations are described. Starting from unitary irreducible representations of the ndimen sional affine WeylHeisenberg group, which are not squareintegrable, one is led to consider systems of coherent states labeled by the elements of quotients of the original group. Such systems can yield a resolution of the identity, and then be used as alternatives to usual wavelet or windowed Fourier analysis. When the quotient space is the phase space of the representation, different embeddings of it into the group provide different descriptions of the phase space. R'esum'e: On d'ecrit des syst`emes d"etats coh'erents engendr'es par des translations, dilatations et rotations en ndimensions. Partant de repr'esentations unitaires irr'eductibles du groupe de WeylHeisenberg affine en dimension n, qui ne sont pas de carr'e integrable, on est naturellement conduit `a consid'erer des syst`emes d"etats coh...
On the decoupling of finite singularities from the question of asymptotic completeness in two body quantum systems
 J. Funct. Anal
, 1976
"... Let V be a multiplication operator, whose negative part, V ( V < 0) obeysA + (1 + c)V>c for some c, c> 0. Let W = Vx where x is the characteristic function of the exterior of a ball. Our main result asserts that the scattering forA + V is complete if and only if that forA + W is complete. Ou ..."
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Cited by 11 (4 self)
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Let V be a multiplication operator, whose negative part, V ( V < 0) obeysA + (1 + c)V>c for some c, c> 0. Let W = Vx where x is the characteristic function of the exterior of a ball. Our main result asserts that the scattering forA + V is complete if and only if that forA + W is complete. Our technical estimates exploit Wiener integrals and the FeynmanKac formula. We also make an application to acoustical scattering. 1.
On the interpolation of injective or projective tensor products of Banach spaces
 J. Funct. Anal
, 1991
"... Abstract: We prove a general result on the factorization of matrixvalued analytic functions. We deduce that if (E0, E1) and (F0, F1) are interpolation pairs with dense intersections, then under some conditions on the spaces E0, E1, F0 and F1, we have [E0 ̂⊗F0, E1 ̂⊗F1]θ = [E0, E1]θ ̂⊗[F0, F1]θ, 0 < ..."
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Cited by 11 (0 self)
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Abstract: We prove a general result on the factorization of matrixvalued analytic functions. We deduce that if (E0, E1) and (F0, F1) are interpolation pairs with dense intersections, then under some conditions on the spaces E0, E1, F0 and F1, we have [E0 ̂⊗F0, E1 ̂⊗F1]θ = [E0, E1]θ ̂⊗[F0, F1]θ, 0 < θ < 1. We find also conditions on the spaces E0, E1, F0 and F1, so that the following holds
The Tresholding Greedy Algorithm, Greedy Bases, and Duality
 CONSTR. APPROX
, 2001
"... Some new conditions that arise naturally in the study of the Thresholding Greedy Algorithm are introduced for bases of Banach spaces. We relate these conditions to best nterm approximation and we study their duality theory. In particular, we obtain a complete duality theory for greedy bases. ..."
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Cited by 11 (6 self)
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Some new conditions that arise naturally in the study of the Thresholding Greedy Algorithm are introduced for bases of Banach spaces. We relate these conditions to best nterm approximation and we study their duality theory. In particular, we obtain a complete duality theory for greedy bases.
Banach Spaces Determined By Their Uniform Structures
 Funct. Anal
, 1996
"... Following results of Bourgain and Gorelik we show that the spaces ` p , 1 ! p ! 1, as well as some related spaces have the following uniqueness property: If X is a Banach space uniformly homeomorphic to one of these spaces then it is linearly isomorphic to the same space. We also prove that if a C(K ..."
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Cited by 10 (3 self)
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Following results of Bourgain and Gorelik we show that the spaces ` p , 1 ! p ! 1, as well as some related spaces have the following uniqueness property: If X is a Banach space uniformly homeomorphic to one of these spaces then it is linearly isomorphic to the same space. We also prove that if a C(K) space is uniformly homeomorphic to c 0 , then it is isomorphic to c 0 . We show also that there are Banach spaces which are uniformly homeomorphic to exactly 2 isomorphically distinct spaces. Subject classification: 46B20, 54Hxx. Keywords: Banach spaces, Uniform homeomorphism, Lipschitz homeomorphism * Erna and Jacob Michael Visiting Professor, The Weizmann Institute, 1994 y Supported in part by NSF DMS 9306376 z Supported in part by the U.S.Israel Binational Science Foundation + Participant, Workshop in Linear Analysis and Probability, Texas A&M University 0. Introduction The first result in the subject we study is the MazurUlam theorem which says that an isometry from one Ban...