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145
Modulation Spaces on Locally Compact Abelian Groups
 UNIVERSITY OF VIENNA
, 1983
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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 79 (47 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.
LittlewoodPaley theory and function spaces with . . .
, 1998
"... Characterizations via convolutions with smooth compactly supported kernels and other distinguished properties of the weighted BesovLipschitz and TriebelLizorkin spaces on IR n with weights that are locally in Ap but may grow or decrease exponentially at infinity are investigated. Squarefunct ..."
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Cited by 34 (1 self)
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Characterizations via convolutions with smooth compactly supported kernels and other distinguished properties of the weighted BesovLipschitz and TriebelLizorkin spaces on IR n with weights that are locally in Ap but may grow or decrease exponentially at infinity are investigated. Squarefunction characterizations of the weighted L p and Hardy spaces with the above class of weights are also obtained. A certain local variant of the Calder'on reproducing formula is constructed and widely used in the proofs.
Stability results on interpolation scales of quasiBanach spaces and applications
 Trans. Amer. Math. Soc
, 1998
"... Abstract. We investigate stability of Fredholm properties on interpolation scales of quasiBanach spaces. This analysis is motivated by problems arising in PDE’s and several applications are presented. In this paper we initiate the study of stability of Fredholm properties of operators on complex in ..."
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Cited by 29 (6 self)
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Abstract. We investigate stability of Fredholm properties on interpolation scales of quasiBanach spaces. This analysis is motivated by problems arising in PDE’s and several applications are presented. In this paper we initiate the study of stability of Fredholm properties of operators on complex interpolation scales of quasiBanach spaces. As such, this is a natural continuation and extension of previous work in the literature (cf., e.g., the articles [32], [35], [7], [8]) which only deals with the case of Banach spaces.
The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems
 GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS. 9
"... We show that given a symmetric convex set K ⊂ Rd, the function t − → γ(etK) is logconcave on R, where γ denotes the standard ddimensional Gaussian measure. We also comment on the extension of this property to unconditional logconcave measures and sets, and on the complex case. 1 ..."
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Cited by 28 (2 self)
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We show that given a symmetric convex set K ⊂ Rd, the function t − → γ(etK) is logconcave on R, where γ denotes the standard ddimensional Gaussian measure. We also comment on the extension of this property to unconditional logconcave measures and sets, and on the complex case. 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 20 (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...
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 19 (10 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.