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.
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.
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.
Matrices and quadrature rules for wavelets
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.
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.
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.
Galerkin-wavelet methods for two-point boundary value problems
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
Last Revised: Mar 12, 1998
© Copyright 1997, 1998 Wei-Chang Shann