2,879 research outputs found
Lectures on Nehari's Theorem on the Polydisk
We give a leisurely proof of a result of Ferguson--Lacey (math.CA/0104144)
and Lacey--Terwelleger (math.CA/0601192) on a Nehari theorem for "little"
Hankel operators on a polydisk. If H_b is a little Hankel operator with symbol
b on product Hardy space we have || H_b || \simeq || b ||_{BMO} where BMO is
the product BMO space identified by Chang and Fefferman. This article begins
with the classical Nehari theorem, and presents the necessary background for
the proof of the extension above. The proof of the extension is an induction on
parameters, with a bootstrapping argument. Some of the more technical details
of the earlier proofs are now seen as consequences of a paraproduct theory.Comment: 35 pages. 65 Reference
Wavelets and Fast Numerical Algorithms
Wavelet based algorithms in numerical analysis are similar to other transform
methods in that vectors and operators are expanded into a basis and the
computations take place in this new system of coordinates. However, due to the
recursive definition of wavelets, their controllable localization in both space
and wave number (time and frequency) domains, and the vanishing moments
property, wavelet based algorithms exhibit new and important properties.
For example, the multiresolution structure of the wavelet expansions brings
about an efficient organization of transformations on a given scale and of
interactions between different neighbouring scales. Moreover, wide classes of
operators which naively would require a full (dense) matrix for their numerical
description, have sparse representations in wavelet bases. For these operators
sparse representations lead to fast numerical algorithms, and thus address a
critical numerical issue.
We note that wavelet based algorithms provide a systematic generalization of
the Fast Multipole Method (FMM) and its descendents.
These topics will be the subject of the lecture. Starting from the notion of
multiresolution analysis, we will consider the so-called non-standard form
(which achieves decoupling among the scales) and the associated fast numerical
algorithms. Examples of non-standard forms of several basic operators (e.g.
derivatives) will be computed explicitly.Comment: 32 pages, uuencoded tar-compressed LaTeX file. Uses epsf.sty (see
`macros'
Martingales, endomorphisms, and covariant systems of operators in Hilbert space
We show that a class of dynamical systems induces an associated operator
system in Hilbert space. The dynamical systems are defined from a fixed
finite-to-one mapping in a compact metric space, and the induced operators form
a covariant system in a Hilbert space of L^2-martingales. Our martingale
construction depends on a prescribed set of transition probabilities, given by
a non-negative function. Our main theorem describes the induced martingale
systems completely. The applications of our theorem include wavelets, the
dynamics defined by iterations of rational functions, and sub-shifts in
symbolic dynamics.
In the theory of wavelets, in the study of subshifts, in the analysis of
Julia sets of rational maps of a complex variable, and, more generally, in the
study of dynamical systems, we are faced with the problem of building a unitary
operator from a mapping r in a compact metric space X. The space X may be a
torus, or the state space of subshift dynamical systems, or a Julia set. While
our motivation derives from some wavelet problems, we have in mind other
applications as well; and the issues involving covariant operator systems may
be of independent interest.Comment: 44 pages, LaTeX2e ("jotart" document class); v2: A few opening
paragraphs were added to the paper; an addition where a bit of the history is
explained, and where some more relevant papers are cited. Corrected a
typographical error in Proposition 8.1. v3: A few minor additions: More
motivation and explanations in the Intro; Remark 3.3 is new; and eleven
relevant references/citations are added; v4: corrected and updated
bibliography; v5: more bibliography updates and change of LaTeX document
clas
Continuous Curvelet Transform: I. Resolution of the Wavefront Set
We discuss a Continuous Curvelet Transform (CCT), a transform f â Îf (a, b, θ) of functions f(x1, x2) on R^2, into a transform domain with continuous scale a > 0, location b â R^2, and orientation θ â [0, 2Ď). The transform is defined by Îf (a, b, θ) = {f, Îłabθ} where
the inner products project f onto analyzing elements called curvelets Îł_(abθ) which are smooth and of rapid decay away from an a by âa rectangle with minor axis pointing in direction θ. We call them curvelets because this anisotropic behavior allows them to âtrackâ the
behavior of singularities along curves. They are continuum scale/space/orientation analogs of the discrete frame of curvelets discussed in Candès and Donoho (2002).
We use the CCT to analyze several objects having singularities at points, along lines, and along smooth curves. These examples show that for fixed (x0, θ0), Îf (a, x0, θ0) decays rapidly as a â 0 if f is smooth near x0, or if the singularity of f at x0 is oriented in a different direction than θ_0. Generalizing these examples, we state general theorems showing that decay properties of
Îf (a, x0, θ0) for fixed (x0, θ0), as a â 0 can precisely identify the wavefront set and the H^m- wavefront set of a distribution. In effect, the wavefront set of a distribution is the closure of the set of (x0, θ0) near which Îf (a, x, θ) is not of rapid decay as a â 0; the H^m-wavefront set is the closure of those points (x0, θ0) where the âdirectional parabolic square functionâ
S^m(x, θ) = ( Ę|Îf (a, x, θ)|^2 ^(da) _a^3+^(2m))^(1/2)
is not locally integrable. The CCT is closely related to a continuous transform used by Hart Smith in his study
of Fourier Integral Operators. Smithâs transform is based on strict affine parabolic scaling of a single mother wavelet, while for the transform we discuss, the generating wavelet changes (slightly) scale by scale. The CCT can also be compared to the FBI (Fourier-Bros-Iagolnitzer) and Wave Packets (Cordoba-Fefferman) transforms. We describe their
similarities and differences in resolving the wavefront set
Fast Isogeometric Boundary Element Method based on Independent Field Approximation
An isogeometric boundary element method for problems in elasticity is
presented, which is based on an independent approximation for the geometry,
traction and displacement field. This enables a flexible choice of refinement
strategies, permits an efficient evaluation of geometry related information, a
mixed collocation scheme which deals with discontinuous tractions along
non-smooth boundaries and a significant reduction of the right hand side of the
system of equations for common boundary conditions. All these benefits are
achieved without any loss of accuracy compared to conventional isogeometric
formulations. The system matrices are approximated by means of hierarchical
matrices to reduce the computational complexity for large scale analysis. For
the required geometrical bisection of the domain, a strategy for the evaluation
of bounding boxes containing the supports of NURBS basis functions is
presented. The versatility and accuracy of the proposed methodology is
demonstrated by convergence studies showing optimal rates and real world
examples in two and three dimensions.Comment: 32 pages, 27 figure
Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States
We resume the recent successes of the grid-based tensor numerical methods and
discuss their prospects in real-space electronic structure calculations. These
methods, based on the low-rank representation of the multidimensional functions
and integral operators, led to entirely grid-based tensor-structured 3D
Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core
Hamiltonian and two-electron integrals (TEI) in complexity using
the rank-structured approximation of basis functions, electron densities and
convolution integral operators all represented on 3D
Cartesian grids. The algorithm for calculating TEI tensor in a form of the
Cholesky decomposition is based on multiple factorizations using algebraic 1D
``density fitting`` scheme. The basis functions are not restricted to separable
Gaussians, since the analytical integration is substituted by high-precision
tensor-structured numerical quadratures. The tensor approaches to
post-Hartree-Fock calculations for the MP2 energy correction and for the
Bethe-Salpeter excited states, based on using low-rank factorizations and the
reduced basis method, were recently introduced. Another direction is related to
the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for
finite lattice-structured systems, where one of the numerical challenges is the
summation of electrostatic potentials of a large number of nuclei. The 3D
grid-based tensor method for calculation of a potential sum on a lattice manifests the linear in computational work, ,
instead of the usual scaling by the Ewald-type approaches
- âŚ