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96
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 74 (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.
L p improving bounds for averages along curves
 Michael Christ, Department of Mathematics, University of California, Berkeley, CA 947203840, USA
"... Abstract. We establish local (L p, L q) mapping properties for averages on curves. The exponents are sharp except for endpoints. 1. ..."
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Cited by 22 (2 self)
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Abstract. We establish local (L p, L q) mapping properties for averages on curves. The exponents are sharp except for endpoints. 1.
Convex Functions On The Heisenberg Group
 Calc. Var. Partial Differential Equations
"... Convex functions in Euclidean space can be characterized as universal viscosity subsolutions of all homogeneous fully nonlinear second order elliptic partial di#erential equations. This is the starting point we have chosen for a theory of convex functions on the Heisenberg group. 1. ..."
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Cited by 17 (1 self)
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Convex functions in Euclidean space can be characterized as universal viscosity subsolutions of all homogeneous fully nonlinear second order elliptic partial di#erential equations. This is the starting point we have chosen for a theory of convex functions on the Heisenberg group. 1.
Regularity Properties of Viscosity Solutions of a NonHörmander Degenerate Equation
 J. Math. Pures Appl
, 2001
"... We study the interior regularity properties of the solutions of a nonlinear degenerate equation arising in mathematical finance. We set the problem in the framework of Hrmander type operators without assuming any hypothesis on the degeneracy of the associated Lie algebra. We prove that the viscosity ..."
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Cited by 16 (11 self)
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We study the interior regularity properties of the solutions of a nonlinear degenerate equation arising in mathematical finance. We set the problem in the framework of Hrmander type operators without assuming any hypothesis on the degeneracy of the associated Lie algebra. We prove that the viscosity solutions are indeed classical solutions. 2001 ditions scientifiques et mdicales Elsevier SAS Keywords: Nonlinear degenerate Kolmogorov equation, Interior regularity, Hrmander operators RSUM.  Nous tudions la rgularit intrieure des solutions de viscosit d'une quation non linaire du second ordre dgnre que l'on rencontre en finance mathmatique. Nous tudions le problme par la thorie des oprateurs de Hrmander sans aucune hypothse sur la dgnerescence de l'algbre de Lie engendre. Nous montrons que la solution de viscosit est une solution classique. 2001 ditions scientifiques et mdicales Elsevier SAS 1.
On the Regularity of Solutions to a Nonlinear Ultraparabolic Equation Arising in Mathematical Finance
 in mathematical finance, Differential Integral Equations 14 (6
, 2001
"... We consider the following nonlinear degenerate parabolic equation which arises in some recent problems of mathematical finance:... ..."
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Cited by 16 (12 self)
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We consider the following nonlinear degenerate parabolic equation which arises in some recent problems of mathematical finance:...
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.
Heat Equations in R
 C. J. Funct. Anal
"... Abstract. Let p: C → R be a subharmonic, nonharmonic polynomial and τ ∈ R a parameter. Define ¯ Zτp = ∂ ∂p + τ ∂¯z ∂¯z, a closed, denselydefined operator on L2 (C). If □τp = ¯ Zτp ¯ Z ∗ τp and τ> 0, we solve the heat equation ∂u + □τpu = 0, u(0, z) = f(z), on (0, ∞) × C. The solution comes via ..."
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Cited by 12 (12 self)
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Abstract. Let p: C → R be a subharmonic, nonharmonic polynomial and τ ∈ R a parameter. Define ¯ Zτp = ∂ ∂p + τ ∂¯z ∂¯z, a closed, denselydefined operator on L2 (C). If □τp = ¯ Zτp ¯ Z ∗ τp and τ> 0, we solve the heat equation ∂u + □τpu = 0, u(0, z) = f(z), on (0, ∞) × C. The solution comes via ∂s the heat semigroup e −s□τp, and we show that u(s, z) = e −s□τp [f](z) = ∫
Implicit function theorem in CarnotCaratheodory spaces
 Comm. in Cont. Math
"... In this paper, we prove an implicit function theorem and we study the regularity of the function implicitly defined. The implicit function theorem had already been proved in homogeneous Lie groups by Franchi, Serapioni and Serra Cassano, while the regularity problem of the function implicitly define ..."
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Cited by 12 (4 self)
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In this paper, we prove an implicit function theorem and we study the regularity of the function implicitly defined. The implicit function theorem had already been proved in homogeneous Lie groups by Franchi, Serapioni and Serra Cassano, while the regularity problem of the function implicitly defined was still open even in the simplest Lie group. Keywords: Carnot–Carathéodory spaces; implicit function theorem.
SubRiemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups
, 2007
"... We solve Gromov’s dimension comparison problem for Hausdorff and box counting dimension on Carnot groups equipped with a CarnotCarathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and CarnotCarathéodory balls, ..."
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Cited by 12 (8 self)
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We solve Gromov’s dimension comparison problem for Hausdorff and box counting dimension on Carnot groups equipped with a CarnotCarathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and CarnotCarathéodory balls, and elements of subRiemannian fractal geometry associated to horizontal selfsimilar iterated function systems on Carnot groups. Inspired by Falconer’s work on almost sure dimensions of Euclidean selfaffine fractals we show that CarnotCarathéodory selfsimilar fractals are almost surely horizontal. As a consequence we obtain explicit dimension formulae for invariant sets of Euclidean iterated function systems of polynomial type. Jet space Carnot groups
SubRiemannian calculus on hypersurfaces in Carnot groups
 Advances in Math. 215
, 2007
"... 2. Carnot groups 7 3. Two basic models 10 4. The subbundle of horizontal planes 13 ..."
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Cited by 12 (1 self)
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2. Carnot groups 7 3. Two basic models 10 4. The subbundle of horizontal planes 13