# Preprints and Technical Reports

l2 is not closed under convolution
With Chien-Chang Yen. Manuscript. Sep 11, 1999.
Abstract. This article is written to faculty. We hope it will be helpful to colleagues who are preparing wavelets or Fourier analysis or DSP related applied math courses for undergraduate students.
Exact Solutions for Daubechies Orthonormal Scaling Coefficients
With Chien-Chang Yen. Technical Report 9704, Department of Mathematics, National Central University, Taiwan. Sep 13, 1997.
Abstract. We review the procedure for the computation of the filter coefficients for the Daubechies' orthonormal scaling functions. It is well known that there are exact solutions for these coefficients of length 2, 4, and 6. Now we can construct the exact solutions for those of length 8 and 10.
Cohen's cycles and orthonormal scaling functions
With Chien-Chang Yen. Technical Report 9703, Department of Mathematics, National Central University, Taiwan. May 12, 1997.
Abstract. There is a set of equivalence conditions for the orthonormality of the scaling functions. Among them, there is the Cohen's cycle condition. We will give more criteria for this condition: If the corresponding filter characteristic function has no zeros in certain closed intervals, then there are finitely many points to be checked for the validity of Cohen's cycle condition. However, these closed intervals cannot be made arbitrarily small.
A scheme to elevate the degree and regularity of biorthogonal wavelets
With Jeng-Nan Tzeng. Preprint on Feb 24, 1997.
Abstract. We will demonstrate a scheme that can elevate orthonormal or biorthogonal wavelets to a new system of biorthogonal wavelet. The approximation degree, that is, the number of vanishing moments, of the elevated biorthogonal wavelet will be higher than the original one. Starting with B-splines and cooperating with the lifting scheme (Sweldens), our elevation scheme can reproduce the biorthogonal wavelets of compact support (Cohen, Daubechies, Feauveau). There is a simple algorithm to elevate from the old filter coefficients to the new filter coefficients. Starting with orthonormal wavelets, the elevated scaling functions are differentiable and the derivatives are nearly orthonormal to their translations. That is, the inner products of the translated derivatives always have the values {-1, 2, -1}. This feature resembles the difference wavelet (Chern, Shann) which may be valuable for solving differential equations.
Quadrature rules needed in Galerkin-wavelets methods
With Chien-Chang Yen. Technical Report 94016, Department of Mathematics, National Central University, Taiwan.
Abstract. In this article we will review several numerical quadrature rules, their theoretical background and implementations, that are needed in a discretization of Galerkin type for differential euqations, using wavelets as basis functions. Material given here will be a summary of results of the recent three years by several groups of researchers in this field, including the author himself.
Quadratures involving polynomials and Daubechies' wavelets
With Chien-Chang Yen. Technical Report 9301, Department of Mathematics, National Central University, Taiwan.
Abstract. Scaling equations are used to derive formulae of quadratures involving polynomials and scaling/wavelet functions with compact supports; in particular, those discovered by Daubechies. It turns out that with a few parameters, which are theoretically exact, these quadratures can be evaluated with algebraic formulae instead of numerical approximations. Those parameters can be obtained with high precision by solving well-conditioned linear systems of equations which involve matrices already seen in the literature of wavelets for other purposes.

Created: Jan 18, 1997
Last Revised: Nov 25, 1999
© Copyright 1999 Wei-Chang Shann

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