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Convergence rates of vector cascade algorithms in L_p

JOURNAL OF APPROXIMATION THEORY 2005年第1期137卷 123-142页

作者： Li, S Zhejiang Univ Dept Math Hangzhou 310027 Zhejiang Peoples R China

We investigate the solutions of vector refinement equations of the form phi = Sigma(alpha is an element of Z alpha) a(alpha)phi(M (.) -alpha), where the vector of functions phi = (phi(1),..., phi(r))(T) is in (L-p(R-s))(r). 1 <= p <= infinity. a =: (a(alpha))(alpha is an element of Zs) is a finitely supported sequence of r x r matrices called the refinement mask, and M is an s x s integer matrix such that lim(n ->infinity) M-n = 0. Associated with the mask a and M is a linear operator Q(a) defined on (L-p(R-s))(r) by Q(a psi) := Sigma(beta is an element of Zs) a(beta)psi(M (.) -beta). The iteration scheme (Q(a)(n)psi)(n=1,2....) is called a cascade algorithm (see [D.R. Chen, R.Q. Jia. S.D. Riemenschneider, Convergence or vector subdivision schemes in Sobolev spaces, Appl. Comput. Harmon. Anal. 12 (2002) 128-149;B. Han, The initial functions in a cascade algorithm, in: D.X. Zhou (Ed.), Proceeding of International Conference of Computational Harmonic Analysis in Hong Kong. 2002;B. Han, R.Q. Jia, Multivariate refinement equations and convergence of subdivision schemes. SIAM J. Math. Anal. 29 (1998) 1177-1199: R.Q. Jia, Subdivision schemes in L-p spaces, Adv. Comput. Math. 3 (1995) 309-341;R.Q. Jia, S.D. Riemenschneider. D.X. Zhou. Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998) 1533-1363: S. Li, Characterization of smoothness of multivariate refinable functions and convergence of cascade algorithms associated with nonhomogeneous refinement equations. Adv. Comput. Math. 20 (2004) 311-331;Q. Sun, Convergence and boundedness of cascade algorithm in Besov space and Triebel-Lizorkin space 1, Adv. Math. (China) 29 (2000) 507-526]). cascade algorithm is an important issue to wavelets analysis and computer graphics. Main results of this paper are related to the convergence and convergence rates of vector cascade algorithm in (L-p(R-s))(r)(1 <= p <= infinity). We give some characterizations on convergence of cascade algorithm and also give estimates on c

关键词： refinement equation cascade algorithm multiple refinable functions convergence rates smoothness

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Direct algorithm for computation of derivatives of the Daubechies basis functions

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APPLIED MATHEMATICS AND COMPUTATION 2005年第2期170卷 1006-1013页

作者： Lin, WB Kovvali, N Carin, L Duke Univ Dept Elect & Comp Engn Durham NC 27708 USA

We extend the direct algorithm for computing the derivatives of the compactly supported Daubechies N-vanishing-moment basis functions. The method yields exact values at dyadic rationals for the nth derivative (0 <= n <= N - 1) of the basis functions, when it exists. Example results are shown for the first derivatives of the basis functions from the Daubechies N-vanishing-moment extremal phase orthonormal family (for N = 3,4, and 5), and the CDF(2, N) spline-based biorthogonal family (for N = 6,8 and 10). (c) 2005 Elsevier Inc. All rights reserved.

关键词： wavelets derivatives Daubechies basis functions cascade algorithm direct algorithm

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Subdivision schemes with nonnegative masks

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MATHEMATICS OF COMPUTATION 2005年第250期74卷 819-839页

作者： Zhou, XL China Jiliang Univ Dept Math Hangzhou Peoples R China Univ Duisburg Essen Inst Math D-47057 Duisburg Germany

The conjecture concerning the characterization of a convergent univariate subdivision algorithm with nonnegative finite mask is confirmed.

关键词： cascade algorithm joint spectral radius nonnegative mask subdivision scheme

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A NOTE ON VECTOR cascade algorithm

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Journal of Computational Mathematics 2002年第4期20卷 363-372页

作者： Qiu-hui Chen Jin-zhao Liu Wen-sheng Zhang Department of Scientific Computing and Computer Applications Zhongshan UniversityGuangzhou510275China Department of Geophysics Peking UniversityBeijing 100871China ICMSEC Academy of Mathematics and Systems SciencesChinese Academy of SciencesBeijing 100080China

The focus of this paper is on the relationship between accuracy of multivariate refinable vector and vector cascade algorithm. We show that, if the vector cascade algorithm (1.5) with isotropic dilation converges to a... 详细信息

关键词： cascade algorithm accuracy symbol refinable vector

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Characterization of smoothness of multivariate refinable functions and convergence of cascade algorithms of nonhomogeneous refinement equations

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2004年第4期20卷 311-331页

作者： Li, S Zhejiang Univ Dept Math Hangzhou 310027 Zhejiang Peoples R China

This paper concerns multivariate homogeneous refinement equations of the form (GRAPHICS) This paper concerns multivariate homogeneous refinement equations of the form (GRAPHICS) where phi=(phi(1),...,phi(r))(T) is the unknown, M is an s x s dilation matrix with m = \detM\, g = (g(1),...,g(r))(T) is a given compactly supported vector-valued function on R-s, and a is a finitely supported refinement mask such that each a(alpha) is an r x r (complex) matrix. In this paper, we characterize the optimal smoothness of a multiple refinable function associated with homogeneous refinement equations in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite-dimensional invariant subspace when M is an isotropic dilation matrix. Nonhomogeneous refinement equations naturally occur in multi-wavelets constructions. Let phi(0) be an initial vector of functions in the Sobolev space (W-2(k)(R-s))(r) (kis an element ofN). The corresponding cascade algorithm is given by (GRAPHICS) We also provide necessary and sufficient conditions for the strong convergence of the cascade algorithm in the Sobolev space (W-2(K) (R-s))(r) (kis an element ofN)) for the case in which M is isotropic.

关键词： refinement equation refinable function smoothness cascade algorithm Sobolev space Lipschitz space transition operator self-affine tile

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Multivariate Refinement Equations and Convergence of cascade algorithms in L_(p)(0

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Acta Mathematica Sinica,English Series 2003年第1期19卷 97-106页

作者： Song LI Department of Mathematices Zhejiang UniversityHangzhou 310027P.R.China Institute of Mathematics Academy of Mathematics and System sciencesChinese Academy of SciencesBeijing 100080P.R.China

We consider the solutions of refinement equations written in the form$$\varphi \left( x \right) = \sum\limits_{\alpha \in \Zopf^s} {a\left( \alpha \right)\varphi \left( {Mx - \alpha } \right) + g\left( x \right),\,\,\,x \in \Ropf^s} $$where the vector of functions } = (}1, ..., }r)T is unknown, g is a given vector of compactly supported functions on A^s, a is a finitely supported sequence of r 2 r matrices called the refinement mask, and M is an s 2 s dilation matrix with m = |detM|. Inhomogeneous refinement equations appear in the construction of multiwavelets and the constructions of wavelets on a finite interval. The cascade algorithm with mask a, g, and dilation M generates a sequence }n, n = 1, 2, ..., by the iterative process$$\varphi _n \left( x \right) = \sum\limits_{\alpha \in \Zopf^s} {a\left( \alpha \right)\varphi _{n - 1} \left(Mx - \alpha \right) + g\left( x \right),\,\,\,x \in \Ropf^s} $$from a starting vector of function }0. We characterize the Lp-convergence (0 < p < 1) of the cascade algorithm in terms of the p-norm joint spectral radius of a collection of linear operators associated with the refinement mask. We also obtain a smoothness property of the solutions of the refinement equations associated with the homogeneous refinement equation.

关键词： Inhomogeneous refinement equation Joint spectral radius cascade algorithm L_(p)(R^(s))(0

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CONVERGENCE OF cascade algorithmS AND SMOOTHNESS OF REFINABLE DISTRIBUTIONS

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Chinese Annals of Mathematics,Series B 2003年第3期24卷 367-386页

作者： SUN QIYU Department of Mathematics, University of Houston, Houston, TX 77204, USA. Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, 119260, Singapore. Univ Houston Dept Math Houston TX 77204 USA Natl Univ Singapore Dept Math Singapore 119260 Singapore

In this paper, the author at first develops a method to study convergence of the cascadealgorithm in a Banach space without stable assumption on the initial (see Theorem 2.1), andthen applies the previous result on the convergence to characterizing compactly supportedrefinable distributions in fractional Sobolev spaces and Holder continuous spaces (see Theorems3.1, 3.3, and 3.4). Finally the author applies the above characterization to choosing appropriateinitial to guarantee the convergence of the cascade algorithm (see Theorem 4.2).

关键词： cascade algorithm cascade operator Refinable distribution Shift-invariant space Linear independent shifts Stable shifts Fractional Sobolev space

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M-channel lifting factorization of perfect reconstruction filter banks and reversible M-band wavelet transforms

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II-EXPRESS BRIEFS 2003年第12期50卷 963-976页

作者： Chen, YJ Amaratunga, KS MIT Dept Civil & Environm Engn Intelligent Engn Syst Lab Wavelet Grp Cambridge MA 02139 USA

An intrinsic M-channel lifting factorization of perfect reconstruction filter banks (PRFBs) is presented as an extension of Sweldens' conventional two-channel lifting scheme. Given a polyphase matrix E(z) of a finite-impulse response (FIR) M-channel PRFB with det(E(z))=z(-K), K is an element of Z, a systematic M-channel lifting factorization is derived based on the Monic Euclidean algorithm. The M-channel lifting structure provides an efficient factorization and implementation;examples include optimizing the factorization for the number of lifting steps, delay elements, and dyadic coefficients. Specialization to paraunitary building blocks enables the design of paraunitary filter banks based on lifting. We show how to achieve reversible, possibly multiplierless, implementations under finite precision, through the unit diagonal scaling property of the Monic Euclidean algorithm. Furthermore, filter-bank regularity of a desired order can be imposed on the lifting structure, and PRFBs with a prescribed admissible scaling filter are conveniently parameterized.

关键词： cascade algorithm discrete cosine transform (DCT) greatest common divisor laurent polynomials lifting factorization M-band wavelets Monic Euclidean algorithm multiplierless approximations perfect reconstruction filter banks (PRFBs) regularity reversible transforms

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Construction of smooth refinable function vectors by cascade algorithms

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2002年第4期40卷 1354-1368页

作者： Chen, DR Beijing Univ Aeronaut & Astronaut Dept Appl Math Beijing 100083 Peoples R China Hubei Inst Nationalities Enshi 44500 Hubei Peoples R China

This paper establishes an equivalent relation between the convergence of a cascade algorithm in Sobolev space and the convergence of an associated cascade algorithm in L-p space. It reduces the convergence in Sobolev space to that in L-p space. On the other hand, by the equivalence we present an algorithm for construction of refinement masks which generate convergent cascade algorithms in Sobolev space. It is very easy to implement the algorithm. Examples are given to illustrate the theory.

关键词： refinable function vector Sobolev space cascade algorithm factorization of mask joint spectral radius

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Convergence of cascade algorithms in Sobolev spaces for perturbed refinement masks

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JOURNAL OF APPROXIMATION THEORY 2002年第2期119卷 133-155页

作者： Chen, DR Plonka, G Univ Duisburg Inst Math D-47048 Duisburg Germany Beijing Univ Aeronaut & Astronaut Dept Appl Math Beijing 100083 Peoples R China Hubei Inst Nationalities Dept Math Enshi 445000 Hubei Peoples R China

In this paper, the convergence of the cascade algorithm in a Sobolev space is considered if the refinement mask is perturbed. It is proved that the cascade algorithm corresponding to a slightly perturbed mask converges. Moreover, the perturbation of the resulting limit function is estimated in terms of that of the masks. (C) 2002 Elsevier Science (USA).

关键词： cascade algorithm Sobolev space joint spectral radius perturbation of refinable functions

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