350 research outputs found
Orthonormal Compactly Supported Wavelets with Optimal Sobolev Regularity
Numerical optimization is used to construct new orthonormal compactly
supported wavelets with Sobolev regularity exponent as high as possible among
those mother wavelets with a fixed support length and a fixed number of
vanishing moments. The increased regularity is obtained by optimizing the
locations of the roots the scaling filter has on the interval (pi/2,\pi). The
results improve those obtained by I. Daubechies [Comm. Pure Appl. Math. 41
(1988), 909-996], H. Volkmer [SIAM J. Math. Anal. 26 (1995), 1075-1087], and P.
G. Lemarie-Rieusset and E. Zahrouni [Appl. Comput. Harmon. Anal. 5 (1998),
92-105].Comment: 18 pages, 8 figure
Variational Approach in Wavelet Framework to Polynomial Approximations of Nonlinear Accelerator Problems
In this paper we present applications of methods from wavelet analysis to
polynomial approximations for a number of accelerator physics problems.
According to variational approach in the general case we have the solution as a
multiresolution (multiscales) expansion in the base of compactly supported
wavelet basis. We give extension of our results to the cases of periodic
orbital particle motion and arbitrary variable coefficients. Then we consider
more flexible variational method which is based on biorthogonal wavelet
approach. Also we consider different variational approach, which is applied to
each scale.Comment: LaTeX2e, aipproc.sty, 21 Page
Shift Unitary Transform for Constructing Two-Dimensional Wavelet Filters
Due to the difficulty for constructing two-dimensional wavelet filters, the commonly used wavelet filters are tensor-product of one-dimensional wavelet filters. In some applications, more perfect reconstruction filters should be provided. In this paper, we introduce a transformation which is referred to as Shift Unitary Transform (SUT) of Conjugate Quadrature Filter (CQF). In terms of this transformation, we propose a parametrization method for constructing two-dimensional orthogonal wavelet filters. It is proved that tensor-product wavelet filters are only special cases of this parametrization method. To show this, we introduce the SUT of one-dimensional CQF and present a complete parametrization of one-dimensional wavelet system. As a result, more ways are provided to randomly generate two-dimensional perfect reconstruction filters
Nonlinear Dynamics of Accelerator via Wavelet Approach
In this paper we present the applications of methods from wavelet analysis to
polynomial approximations for a number of accelerator physics problems. In the
general case we have the solution as a multiresolution expansion in the base of
compactly supported wavelet basis. The solution is parametrized by the
solutions of two reduced algebraical problems, one is nonlinear and the second
is some linear problem, which is obtained from one of the next wavelet
constructions: Fast Wavelet Transform, Stationary Subdivision Schemes, the
method of Connection Coefficients. According to the orbit method and by using
construction from the geometric quantization theory we construct the symplectic
and Poisson structures associated with generalized wavelets by using
metaplectic structure. We consider wavelet approach to the calculations of
Melnikov functions in the theory of homoclinic chaos in perturbed Hamiltonian
systems and for parametrization of Arnold-Weinstein curves in Floer variational
approach.Comment: 16 pages, no figures, LaTeX2e, aipproc.sty, aipproc.cl
Wavelets and Wavelet Packets on Quantum Computers
We show how periodized wavelet packet transforms and periodized wavelet
transforms can be implemented on a quantum computer. Surprisingly, we find that
the implementation of wavelet packet transforms is less costly than the
implementation of wavelet transforms on a quantum computer.Comment: 11 pages, 10 postscript figure, to appear in Proc. of Wavelet
Applications in Signal and Image Processing VI
Parabolic Molecules
Anisotropic decompositions using representation systems based on parabolic
scaling such as curvelets or shearlets have recently attracted significantly
increased attention due to the fact that they were shown to provide optimally
sparse approximations of functions exhibiting singularities on lower
dimensional embedded manifolds. The literature now contains various direct
proofs of this fact and of related sparse approximation results. However, it
seems quite cumbersome to prove such a canon of results for each system
separately, while many of the systems exhibit certain similarities.
In this paper, with the introduction of the notion of {\em parabolic
molecules}, we aim to provide a comprehensive framework which includes
customarily employed representation systems based on parabolic scaling such as
curvelets and shearlets. It is shown that pairs of parabolic molecules have the
fundamental property to be almost orthogonal in a particular sense. This result
is then applied to analyze parabolic molecules with respect to their ability to
sparsely approximate data governed by anisotropic features. For this, the
concept of {\em sparsity equivalence} is introduced which is shown to allow the
identification of a large class of parabolic molecules providing the same
sparse approximation results as curvelets and shearlets. Finally, as another
application, smoothness spaces associated with parabolic molecules are
introduced providing a general theoretical approach which even leads to novel
results for, for instance, compactly supported shearlets
Convergence of the cascade algorithm at irregular scaling functions
The spectral properties of the Ruelle transfer operator which arises from a
given polynomial wavelet filter are related to the convergence question for the
cascade algorithm for approximation of the corresponding wavelet scaling
function.Comment: AMS-LaTeX; 38 pages, 10 figures comprising 42 EPS diagrams; some
diagrams are bitmapped at 75 dots per inch; for full-resolution bitmaps see
ftp://ftp.math.uiowa.edu/pub/jorgen/convcasc
Classical and Quantum Ensembles via Multiresolution. II. Wigner Ensembles
We present the application of the variational-wavelet analysis to the
analysis of quantum ensembles in Wigner framework. (Naive) deformation
quantization, the multiresolution representations and the variational approach
are the key points. We construct the solutions of Wigner-like equations via the
multiscale expansions in the generalized coherent states or high-localized
nonlinear eigenmodes in the base of the compactly supported wavelets and the
wavelet packets. We demonstrate the appearance of (stable) localized patterns
(waveletons) and consider entanglement and decoherence as possible
applications.Comment: 5 pages, 2 figures, espcrc2.sty, Presented at IX International
Workshop on Advanced Computing and Analysis Techniques in Physics Research,
Section III "Simulations and Computations in Theoretical Physics and
Phenomenology", ACAT 2003, December, 2003, KEK, Tsukub
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