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112
Diffusion Wavelets
, 2004
"... We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their ..."
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Cited by 72 (12 self)
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We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. In this case our construction can be viewed as a farreaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the nonstandard wavelet representation of CalderónZygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical LittlewoodPaley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.
Gaussian estimates for Markov chains and random walks on groups
 Ann. Probab
, 1993
"... JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JS ..."
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Cited by 37 (2 self)
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.
Regularity Of Irregular Subdivision
, 1998
"... . We study the smoothness of the limit function for one dimensional unequally spaced interpolating subdivision schemes. The new grid points introduced at every level can lie in irregularly spaced locations between old, adjacent grid points and not only midway as is usually the case. For the natural ..."
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Cited by 30 (5 self)
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. We study the smoothness of the limit function for one dimensional unequally spaced interpolating subdivision schemes. The new grid points introduced at every level can lie in irregularly spaced locations between old, adjacent grid points and not only midway as is usually the case. For the natural generalization of the four point scheme introduced by Dubuc and Dyn, Levin, and Gregory, we show that, under some geometric restrictions, the limit function is always C 1 ; under slightly stronger restrictions we show that the limit function is almost C 2 , the same regularity as in the regularly spaced case. 1. Introduction Subdivision is a powerful mechanism for the construction of smooth curves and surfaces. The main idea behind subdivision is to iterate upsampling and local averaging to build complex geometrical shapes. Originally such schemes were studied in the context of corner cutting [13, 5] as well as for building piecewise polynomial curves, e.g., the de Casteljau algorithm f...
GalerkinWavelet Methods For TwoPoint Boundary Value Problems
 NUMER. MATH
, 1992
"... Antiderivatives of wavelets are used for the numerical solution of differential equations. Optimal error estimates are obtained in the applications to twopoint boundary value problems of second order. The orthogonal property of the wavelets is used to construct efficient iterative methods for the ..."
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Cited by 29 (6 self)
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Antiderivatives of wavelets are used for the numerical solution of differential equations. Optimal error estimates are obtained in the applications to twopoint boundary value problems of second order. The orthogonal property of the wavelets is used to construct efficient iterative methods for the solution of the resultant linear algebraic systems. Numerical examples are given.
Riesz transform, Gaussian bounds and the method of wave equation
 Math. Z
"... Abstract. For an abstract selfadjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α> 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We al ..."
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Cited by 20 (1 self)
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Abstract. For an abstract selfadjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α> 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature. As an application of the obtained results we prove boundedness of the Riesz transform on L p for all p ∈ (1, 2] for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on L p of the LaplaceBeltrami operator on Riemannian manifolds for p> 2. 1.
Wiener’s lemma for infinite matrices
 Trans. Amer. Math. Soc
, 2006
"... Abstract. The classical Wiener lemma and its various generalizations are important and have numerous applications in numerical analysis, wavelet theory, frame theory, and sampling theory. There are many different equivalent formulations for the classical Wiener lemma, with an equivalent formulation ..."
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Cited by 18 (11 self)
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Abstract. The classical Wiener lemma and its various generalizations are important and have numerous applications in numerical analysis, wavelet theory, frame theory, and sampling theory. There are many different equivalent formulations for the classical Wiener lemma, with an equivalent formulation suitable for our generalization involving commutative algebra of infinite matrices W: = {(a(j − j ′)) j,j ′ ∈Zd: � j∈Zd a(j)  < ∞}. In the study of spline approximation, (diffusion) wavelets and affine frames, Gabor frames on nonuniform grid, and nonuniform sampling and reconstruction, the associated algebras of infinite matrices are extremely noncommutative, but we expect those noncommutative algebras to have a similar property to Wiener’s lemma for the commutative algebra W. In this paper, we consider two noncommutative algebras of infinite matrices, the Schur class and the Sjöstrand class, and establish Wiener’s lemmas for those matrix algebras. 1.
Frames in spaces with finite rate of innovations
 Adv. Comput. Math
"... Abstract. Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space Vq(Φ, Λ) modelling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applic ..."
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Cited by 14 (12 self)
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Abstract. Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space Vq(Φ, Λ) modelling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applications, cellular radio and ultra wideband communication. In particular, the space Vq(Φ, Λ) is generated by a family of welllocalized molecules Φ of similar size located on a relativelyseparated set Λ using ℓ q coefficients, and hence is locally finitelygenerated. Moreover that space Vq(Φ, Λ) includes finitelygenerated shiftinvariant spaces, spaces of nonuniform splines, and the twisted shiftinvariant space in Gabor (Wilson) system as its special cases. Use the welllocalization property of the generator Φ, we show that if the generator Φ is a frame for the space V2(Φ, Λ) and has polynomial (subexponential) decay, then its canonical dual (tight) frame has the same polynomial (subexponential) decay. We apply the above result about the canonical dual frame to the study of the Banach frame property of the generator Φ for the space Vq(Φ, Λ) with q = 2, and of the polynomial (subexponential) decay property of the mask associated with a refinable function that has polynomial (subexponential) decay. Advances in Computational Mathematics, to appear 1.
NONSMOOTH CALCULUS
"... Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singu ..."
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Cited by 12 (0 self)
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Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singular spaces where calculus is possible parallels the classical development from smooth functions to functions with weak or generalized derivatives. Moreover, there is a new way of looking at the classical geometric theory of Sobolev functions that is useful in more general contexts. 1.
Continuous wavelets and frames on stratified Lie groups I
 Journal of Fourier Analysis and Applications
, 2006
"... Let G be a stratified Lie group and L be the subLaplacian on G. Let 0 ̸ = f ∈ S(R +). We show that Lf(L)δ, the distribution kernel of the operator Lf(L), is an admissible function on G. We also show that, if ξf(ξ) satisfies Daubechies ’ criterion, then Lf(L)δ generates a frame for any sufficiently ..."
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Cited by 10 (7 self)
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Let G be a stratified Lie group and L be the subLaplacian on G. Let 0 ̸ = f ∈ S(R +). We show that Lf(L)δ, the distribution kernel of the operator Lf(L), is an admissible function on G. We also show that, if ξf(ξ) satisfies Daubechies ’ criterion, then Lf(L)δ generates a frame for any sufficiently fine lattice subgroup of G.
Gatto Boundedness properties of fractional integral operators associated to non–doubling measures, preprint 2002
"... Abstract. The main purpose of this paper is to investigate the behaviour of fractional integral operators associated to a measure on a metric space satisfying just a mild growth condition, namely that the measure of each ball is controlled by a fixed power of its radius. This allows, in particular, ..."
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Cited by 10 (2 self)
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Abstract. The main purpose of this paper is to investigate the behaviour of fractional integral operators associated to a measure on a metric space satisfying just a mild growth condition, namely that the measure of each ball is controlled by a fixed power of its radius. This allows, in particular, non–doubling measures. It turns out that this condition is enough to build up a theory that contains the classical results based upon the Lebesgue measure on euclidean space and their known extensions for doubling measures. We start by analyzing the images of the Lebesgue spaces associated to the measure. The Lipschitz spaces, defined in terms of the metric, play a basic role too. For a euclidean space equipped with one of these measures, we also consider the socalled “regular”BMO space introduced by X. Tolsa. We show that it contains the image of a Lebesgue space in the appropriate limit case and also that the image of the space “regular”BMO is contained in the adequate Lipschitz space. 1.