75,804 research outputs found
Application of wavelets to singular integral scattering equations
The use of orthonormal wavelet basis functions for solving singular integral
scattering equations is investigated. It is shown that these basis functions
lead to sparse matrix equations which can be solved by iterative techniques.
The scaling properties of wavelets are used to derive an efficient method for
evaluating the singular integrals. The accuracy and efficiency of the wavelet
transforms is demonstrated by solving the two-body T-matrix equation without
partial wave projection. The resulting matrix equation which is characteristic
of multiparticle integral scattering equations is found to provide an efficient
method for obtaining accurate approximate solutions to the integral equation.
These results indicate that wavelet transforms may provide a useful tool for
studying few-body systems.Comment: 11 pages, 4 figure
Nonhomogeneous Wavelet Systems in High Dimensions
It is of interest to study a wavelet system with a minimum number of
generators. It has been showed by X. Dai, D. R. Larson, and D. M. Speegle in
[11] that for any real-valued expansive matrix M, a homogeneous
orthonormal M-wavelet basis can be generated by a single wavelet function. On
the other hand, it has been demonstrated in [21] that nonhomogeneous wavelet
systems, though much less studied in the literature, play a fundamental role in
wavelet analysis and naturally link many aspects of wavelet analysis together.
In this paper, we are interested in nonhomogeneous wavelet systems in high
dimensions with a minimum number of generators. As we shall see in this paper,
a nonhomogeneous wavelet system naturally leads to a homogeneous wavelet system
with almost all properties preserved. We also show that a nonredundant
nonhomogeneous wavelet system is naturally connected to refinable structures
and has a fixed number of wavelet generators. Consequently, it is often
impossible for a nonhomogeneous orthonormal wavelet basis to have a single
wavelet generator. However, for redundant nonhomogeneous wavelet systems, we
show that for any real-valued expansive matrix M, we can always
construct a nonhomogeneous smooth tight M-wavelet frame in with a
single wavelet generator whose Fourier transform is a compactly supported
function. Moreover, such nonhomogeneous tight wavelet frames are
associated with filter banks and can be modified to achieve directionality in
high dimensions. Our analysis of nonhomogeneous wavelet systems employs a
notion of frequency-based nonhomogeneous wavelet systems in the distribution
space. Such a notion allows us to separate the perfect reconstruction property
of a wavelet system from its stability in function spaces
Oversampling of wavelet frames for real dilations
We generalize the Second Oversampling Theorem for wavelet frames and dual
wavelet frames from the setting of integer dilations to real dilations. We also
study the relationship between dilation matrix oversampling of semi-orthogonal
Parseval wavelet frames and the additional shift invariance gain of the core
subspace.Comment: Journal of London Mathematical Society, published online March 13,
2012 (to appear in print
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