5,336 research outputs found
Compactly Supported Wavelets Derived From Legendre Polynomials: Spherical Harmonic Wavelets
A new family of wavelets is introduced, which is associated with Legendre
polynomials. These wavelets, termed spherical harmonic or Legendre wavelets,
possess compact support. The method for the wavelet construction is derived
from the association of ordinary second order differential equations with
multiresolution filters. The low-pass filter associated with Legendre
multiresolution analysis is a linear phase finite impulse response filter
(FIR).Comment: 6 pages, 6 figures, 1 table In: Computational Methods in Circuits and
Systems Applications, WSEAS press, pp.211-215, 2003. ISBN: 960-8052-88-
Non-equispaced B-spline wavelets
This paper has three main contributions. The first is the construction of
wavelet transforms from B-spline scaling functions defined on a grid of
non-equispaced knots. The new construction extends the equispaced,
biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new
construction is based on the factorisation of wavelet transforms into lifting
steps. The second and third contributions are new insights on how to use these
and other wavelets in statistical applications. The second contribution is
related to the bias of a wavelet representation. It is investigated how the
fine scaling coefficients should be derived from the observations. In the
context of equispaced data, it is common practice to simply take the
observations as fine scale coefficients. It is argued in this paper that this
is not acceptable for non-interpolating wavelets on non-equidistant data.
Finally, the third contribution is the study of the variance in a
non-orthogonal wavelet transform in a new framework, replacing the numerical
condition as a measure for non-orthogonality. By controlling the variances of
the reconstruction from the wavelet coefficients, the new framework allows us
to design wavelet transforms on irregular point sets with a focus on their use
for smoothing or other applications in statistics.Comment: 42 pages, 2 figure
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
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