Results 1  10
of
88
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 ..."
Abstract

Cited by 72 (12 self)
 Add to MetaCart
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.
Noncommutative martingale inequalities
, 1997
"... We prove the analogue of the classical BurkholderGundy inequalites for noncommutative martingales. As applications we give a characterization for an ItoClifford integral to be an Lpmartingale via its integrand, and then extend the ItoClifford integral theory in L2, developed by Barnett, Streater ..."
Abstract

Cited by 41 (9 self)
 Add to MetaCart
We prove the analogue of the classical BurkholderGundy inequalites for noncommutative martingales. As applications we give a characterization for an ItoClifford integral to be an Lpmartingale via its integrand, and then extend the ItoClifford integral theory in L2, developed by Barnett, Streater and Wilde, to Lp for all 1 < p < ∞. We include an appendix on the noncommutative analogue of the classical Fefferman duality between H¹ and BMO.
Uniformly elliptic operators on Riemannian manifolds
 J. Diff. Geom
, 1992
"... Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiiso ..."
Abstract

Cited by 33 (2 self)
 Add to MetaCart
Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiisometric to g. We first prove some Poincare and Sobolev inequalities on geodesic balls. Then we use Moser's iteration to obtain Harnack inequalities. Gaussian estimates, uniqueness theorems, and other applications are also discussed. These results involve local or global lower bound hypotheses on the Ricci curvature of g. Some of them are new even when applied to the Laplace operator of (M, g). 1.
Convolution operator and maximal function for Dunkl transform
 J. Anal. Math
"... Abstract. For a family of weight functions, hκ, invariant under a finite reflection group on R d, analysis related to the Dunkl transform is carried out for the weighted L p spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inver ..."
Abstract

Cited by 23 (2 self)
 Add to MetaCart
Abstract. For a family of weight functions, hκ, invariant under a finite reflection group on R d, analysis related to the Dunkl transform is carried out for the weighted L p spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the BochnerRiesz means. We also define a maximal function and use it to prove the almost everywhere convergence. 1.
The Construction Of Single Wavelets In DDimensions
 J. GEOM. ANAL
, 1999
"... Sets K in ddimensional Euclidean space are constructed with the property that the inverse Fourier transform of the characteristic function 1K is a single dyadic orthonormal wavelet. The construction is characterized by its generality in terms of a quantitative iterative procedure, by its computat ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
Sets K in ddimensional Euclidean space are constructed with the property that the inverse Fourier transform of the characteristic function 1K is a single dyadic orthonormal wavelet. The construction is characterized by its generality in terms of a quantitative iterative procedure, by its computational implementation, and by its simplicity. The general case in which the inverse Fourier transforms of the characteristic functions 1 K 1 ; : : : ; 1 K L are a family of orthonormal wavelets is treated in [Leo99].
Angular regularity and Strichartz estimates for the wave equation
, 2005
"... We prove here essentially sharp L q (L r) linear and bilinear estimates for the wave equations on Minkowski space where we assume the initial data possesses additional regularity with respect to fractional powers of the angular momentum operators Ωij: = x i ∂j − x j ∂i. In this setting, the range ..."
Abstract

Cited by 16 (1 self)
 Add to MetaCart
We prove here essentially sharp L q (L r) linear and bilinear estimates for the wave equations on Minkowski space where we assume the initial data possesses additional regularity with respect to fractional powers of the angular momentum operators Ωij: = x i ∂j − x j ∂i. In this setting, the range of exponents (q, r) vastly improves over what is available for the wave equations based on translation invariant derivatives of the initial data, or uniform decay of the solution.
Boundary Regularity for the Ricci Equation, Geometric Convergence, and Gel'fand's Inverse Boundary Problem
"... Abstract This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The secon ..."
Abstract

Cited by 14 (13 self)
 Add to MetaCart
Abstract This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional stability of an inverse problem proposed by Gel'fand, making essential use of the results of the first two parts. 1.
Causal geometry of EinsteinVacuum spacetimes with finite curvature flux
 Invent. Math
"... Abstract. One of the central difficulties of settling the L 2bounded curvature conjecture for the EinsteinVacuum equations is to be able to control the causal structure of spacetimes with such limited regularity. In this paper we show how to circumvent this difficulty by showing that the geometry ..."
Abstract

Cited by 14 (7 self)
 Add to MetaCart
Abstract. One of the central difficulties of settling the L 2bounded curvature conjecture for the EinsteinVacuum equations is to be able to control the causal structure of spacetimes with such limited regularity. In this paper we show how to circumvent this difficulty by showing that the geometry of null hypersurfaces of EnsteinVacuum spacetimes can be controlled in terms of initial data and the total curvature flux through the hypersurface.
A geometric approach to the LittlewoodPaley theory
 Geom. Funct. Anal
"... Abstract. We develop a geometric invariant LittlewoodPaley theory for arbitrary tensors on a compact 2 dimensional manifold. We show that all the important features of the classical LP theory survive with estimates which depend only on very limited regularity assumptions on the metric. We give inva ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
Abstract. We develop a geometric invariant LittlewoodPaley theory for arbitrary tensors on a compact 2 dimensional manifold. We show that all the important features of the classical LP theory survive with estimates which depend only on very limited regularity assumptions on the metric. We give invariant descriptions of Sobolev and Besov spaces and prove some sharp product inequalities. This theory has being developed in connection to the work of the authors on the geometry of null hypersurfaces with a finite curvature flux condition, see [KlRodn1], [KlRodn2]. We are confident however that it can be applied, and extended, to many different situations. 1. introduction In its simplest manifestation LittlewoodPaley theory is a systematic method to understand various properties of functions f, defined on Rn, by decomposing them in infinite dyadic sums f = ∑ k∈Z fk, with frequency localized components fk, i.e. ̂fk(ξ) = 0 for all values of ξ outside the annulus 2k−1 ≤ ξ  ≤ 2k+1. Such a decomposition can be easily achieved by choosing a test function χ = χ(ξ) in Fourier ≤ ξ  ≤ 2, and such that, for all ξ ̸ = 0, k∈Z χ(2−kξ) = 1. space, supported in 1 2 Then set ̂ fk(ξ) = χ(2 k ξ) ˆ f(ξ) or, in physical space, Pkf = fk = mk ∗ f where mk(x) = 2 nk m(2 k x) and m(x) the inverse Fourier transform of χ. The operators Pk are called cutoff operators or, improperly, LP projections. We denote PJ = ∑ k∈J Pk for all intervals J ⊂ Z. The following properties of these LP projections are very easy to verify and lie at the heart of the classical LP theory:
Group Invariant Scattering
, 2011
"... Pattern classification often requires using translation invariant representations, which are stable and hence Lipschitz continuous to deformations. A Fourier transform does not provide such Lipschitz stability. Scattering operators are obtained by iterating on wavelet transforms and modulus operator ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
Pattern classification often requires using translation invariant representations, which are stable and hence Lipschitz continuous to deformations. A Fourier transform does not provide such Lipschitz stability. Scattering operators are obtained by iterating on wavelet transforms and modulus operators. The resulting representation is proved to be translation invariant and Lipschitz continuous to deformations, up to a log term. It is computed with a nonlinear convolution network, which scatters functions along an infinite set of paths. Invariance to the action of any compact Lie subgroup of GL(R d) is obtained with a combined scattering, which iterates over wavelet transforms defined on this group. Scattering representations yield new metrics on stationary processes, which are stable to random deformations. 1