Publications
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On the Exact Values of Orthonormal Scaling Coefficients
of Length 8 and 10
- With Mr C-C Yen, published in
Applied and Computational Harmonic Analysis,
6(1999), 109--112.
- Abstract.
We show the exact values of the scaling coefficients of length eight and ten
for Daubechies' orthonormal scaling functions.
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The Levering Scheme of Biorthogonal Wavelets
- With Mr Jen-Nan Tzeng, published in
Proceedings of the International Wavelets Conference ``Wavelets and
Multiscale Methods'', April 13--17, 1998, Tangier, Morocco.
- Abstract.
We present a scheme that will lever orthonormal
or biorthogonal wavelets to a new system of biorthogonal wavelets.
If we start with orthonormal wavelets, the raised scaling functions
and wavelets are compactly supported and are differentiable.
The derivatives of the raised biorthogonal scaling/wavelets
forms an almost orthonormal system.
If we start with B-splines and cooperating with the lifting scheme
of Sweldens, our levering scheme can reproduce all of those biorthogonal
wavelets of compact support by Cohen, Daubechies and Feauveau.
There is a simple algorithm to construct from the old filter coefficients
to the new filter coefficients.
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On the tau-cycle condition
- With Mr C-C Yen, published in
Applied and Computational Harmonic Analysis,
5(1998), 370--373.
- Abstract.
There is a set of equivalence conditions
for the orthonormality of the compactly supported scaling functions.
Among them, there is the Cohen's tau-cycle condition.
In order to answer the question that whether it is enough to check this
condition by a finite number of points,
we study the tau-cycles in more detail.
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Matrices and quadrature rules for wavelets
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With Chien-Chang Yen, published in
Taiwanese Journal of Mathematics
2(1998), 435--446.
- Abstract.
Using the scaling equations, quadratures involving polynomials and
scaling/wavelet functions can be evaluated by linear algebraic
equations (which are theoretically exact) instead of numerical
approximations. We study two matrices which are derived from these
kinds of quadratures. These particular matrices are also seen in the
literature of wavelets for other purposes.
Afternote. This article was appreciated by many practitioners,
but it was not much appreciated by mathematicians. I had difficulties
publishing this article for almost four years. But its electronic
version was downloaded and used by many engineers and scientists.
For instance, Ingo D. Rullhusen from Germany wrote to me on Dec 14, 2000:
Before we have found your article, we have calculated the function
multiplying operator in this way, too, but only less accuracy were
reached with a high demand on sampling points. Now, the analytical
calculatable correlation coefficients "L_{i,j}" are doing this job with
high accuracy and only few sampling points.
This success yields the demand to eliminate the numerical calculation of
the last operator, too.
I am confident that the results in this article are useful. But it is
still very glad to know that they have been actually applied to
the industry.
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About wavelets on the interval
- With Miss J Cheng, M.S., published in
Proceedings of the International Mathematics Conference 1994,
World Scientific (1996), 49-64.
- Abstract.
Wavelets on the closed interval are constructed step by step
from the Daubechies' wavelets with compact supports.
We follow mostly the guidelines by Andersson, Hall, Jawerth and Peters,
with modifications on the construction of ``edge'' wavelets.
Then we construct the filter operators discussed by Cohen, Daubechies
and Vial.
A filtering process is suggested for two-dimensional discrete
transforms based on wavelets on the interval.
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Cyclic reduction method and difference wavelets
- With Professor I-L Chern, published in
Proceedings of the workshop on computational sciences 1994,
National Taiwan University (1994), 25-29.
- Abstract.
The study of a family of direct methods for solving sparse banded
linear systems motivated by multiresolution decomposition gives a new
look of the classical cyclic reduction method. From this point of view
we derive a family of difference wavelets associated with difference
operatiors, which forms a generalized multiresolution analysis in a
different sense of measure.
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Galerkin-wavelet methods for two-point boundary value problems
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With Professor J-C Xu, published in
Numerische Mathematik,
63 (1992), 123-144.
- Abstract.
Anti-derivatives of wavelets are used for the numerical solution of
differential equations. Optimal error estimates are obtained in the
applications to two-point boundary value problems of second order. The
orthogonal property of the wavelets is used to construct efficient
iterative methods for the solution of the resultant linear algebraic
systems. Numerical examples are given.
Created: Jan 18, 1997