作者:
Li, SZhejiang 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...
详细信息
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
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 <=...
详细信息
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.
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
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 vector-valued function with regularity, then the initial function must satisfy the Strang-Fix conditions.
作者:
Li, SZhejiang 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...
详细信息
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.
We consider the solutions of refinement equations written in the form$$\varphi \left( x \right) = \sum\limits_{\alpha \in \Zopf^s} {a\left( \alpha \right)\varphi \left( {Mx - \alpha } \right) + g\left( x \right),\,\,\...
详细信息
We consider the solutions of refinement equations written in the form$$\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.
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 th...
详细信息
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).
作者:
Chen, YJAmaratunga, KSMIT
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 fin...
详细信息
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.
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 ...
详细信息
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.
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 converge...
详细信息
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).
暂无评论