ON PARAMETRIC FACTORIZATION OF BI－ORTHOGONAL LAURENT POLYNOMIAL WAVELET FILTERS

In this paper ,we study the factorization of bi-orthogonal Laurent polynomial wavelet matrices with degree one into simple blocks.A conjecture about advanced factorziation is given.     

On parametric factorization of bi-orthogonal Laurent polynomial wavelet filters

In this paper, we study the factorization of bi-orthogonal Laurent polynomial wavelet matrices with degree one into simple blocks. A conjecture about advanced factorination is given.     

Spectral factorization of Laurent polynomials

We analyse the performance of five numerical methods for factoring a Laurent polynomial, which is positive on the unit circle, as the modulus squared of a real algebraic polynomial. It is found that there is a wide disparity between the methods, and all but one of the methods are significantly influenced by the variation in magnitude of the coefficients of the Laurent polynomial, by the closeness of the zeros of this polynomial to the unit circle, and by the spacing of these zeros. This revised version was published online in August 2006 with corrections to the Cover Date.     

Lubin-Hensel factorization for Laurent series

Summary Let K be a complete ultrametric algebraically closed field. Let D be a bounded closed strongly infraconnected set in K with no T-filter, and let H(D) be the Banach algebra of the analytic elements in D. Let r, r be functions from D toR with bounds a, b such that 0 (D,r,r) be the Banach algebra of the Laurent series with coefficients a_s in H(D) such that , provided with a suitable norm. In (D, r, r) we give a kind of Hensel Factorization for series whose dominating coefficients at r(x) and at r(x) conserve the same rank. We take advantage of this method to correcting a mistake that happened in our previous article on the Hensel Factorization for Taylor series.And Erratum to «Maximum principle for analytic elements and Lubin-Hensel's Theorem inH(D)Y»,135, pp. 265–278 of this Journal.     

On parametric polynomial circle approximation

In the paper, the uniform approximation of a circle arc (or a whole circle) by a parametric polynomial curve is considered. The approximant is obtained in a closed form. It depends on a parameter that should satisfy a particular equation, and it takes only a couple of tangent method steps to compute it. For low degree curves, the parameter is provided exactly. The distance between a circle arc and its approximant asymptotically decreases faster than exponentially as a function of polynomial degree. Additionally, it is shown that the approximant could be applied for a fast evaluation of trigonometric functions too.     

Distributive skew Laurent polynomial rings

We describe skew Laurent polynomial rings that are right distributive.     

Inverse polynomial expansions of Laurent series

An algorithm is introduced, and shown to lead to various unique series expansions of formal Laurent series, as the sums of reciprocals of polynomials. The degrees of approximation by the rational functions which are the partial sums of these series are investigated. The types of series corresponding to rational functions themselves are also characterized.     

Radicals in skew polynomial and skew Laurent polynomial rings

We provide a general procedure for characterizing radical-like functions of skew polynomial and skew Laurent polynomial rings under grading hypotheses. In particular, we are able to completely characterize the Wedderburn and Levitzki radicals of skew polynomial and skew Laurent polynomial rings in terms of ideals in the coefficient ring. We also introduce the T-nilpotent radideals, and perform similar characterizations.     

Sparse bivariate polynomial factorization

Motivated by Sasaki’s work on the extended Hensel construction for solving multivariate algebraic equations, we present a generalized Hensel lifting, which takes advantage of sparsity, for factoring bivariate polynomial over the rational number field. Another feature of the factorization algorithm presented in this article is a new recombination method, which can solve the extraneous factor problem before lifting based on numerical linear algebra. Both theoretical analysis and experimental data show that the algorithm is efficient, especially for sparse bivariate polynomials.