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Smoothness of multiple refinable functions and multiple wavelets

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 1999年第1期21卷 1-28页

作者： Jia, RQ Riemenschneider, SD Zhou, DX Univ Alberta Dept Math Sci Edmonton AB T6G 2G1 Canada City Univ Hong Kong Dept Math Kowloon Peoples R China

We consider the smoothness of solutions of a system of refinement equations written in the form phi = Sigma(alpha is an element of z) a(alpha)phi(2.-alpha), where the vector of functions phi = (phi(1),...,phi(r))(T) is in (L-p( R))(r) and a is a finitely supported sequence of r x r matrices called the refinement mask. We use the generalized Lipschitz space Lip*(nu, L-p(R)), nu >0, to measure smoothness of a given function. Our method is to relate the optimal smoothness, nu(p)(phi), to the p-norm joint spectral radius of the block matrices A(epsilon), epsilon = 0,1, given by A(epsilon) = (a(epsilon + 2 alpha - beta))(alpha,beta), when restricted to a certain finite dimensional common invariant subspace V. Denoting the p-norm joint spectral radius by rho(p)(A(0)\(V), A(1)\(V)), we show that nu(p)(phi) greater than or equal to 1/p - log(2)rho(p)(A(0)\(V), A(1)\(V)) with equality when the shifts of phi(1),...,phi(r) are stable and the invariant subspace is generated by certain vectors induced by difference operators of sufficiently high order. This allows an effective use of matrix theory. Also the computational implementation of our method is simple. When p = 2, the optimal smoothness is also given in terms of the spectral radius of the transition matrix associated with the refinement mask. To illustrate the theory, we give a detailed analysis of two examples where the optimal smoothness can be given explicitly. We also apply our methods to the smoothness analysis of multiple wavelets. These examples clearly demonstrate the applicability and practical power of our approach.

关键词： refinement equations multiple refinable functions multiple wavelets vector subdivision schemes joint spectral radii transition operators

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Convergence of vector subdivision schemes in Sobolev spaces

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APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 2002年第1期12卷 128-149页

作者： Chen, DR Jia, RQ Riemenschneider, SD Beijing Univ Aeronaut & Astronaut Dept Appl Math Beijing 100083 Peoples R China Univ Alberta Dept Math Sci Edmonton AB T6G 2G1 Canada W Virginia Univ Dept Math Morgantown WV 26506 USA

In this paper we consider functional equations of the form phi = Sigma(alphais an element ofZs) a(alpha)phi(M-.-alpha). where phi = (phi(1), . . . . phi(r)()T) is an r x 1 vector of functions on the s-dimensional Euclidean space, a(alpha), alpha is an element of Z(S), is a finitely supported sequence of r x r complex matrices, and M is an s x s isotropic integer matrix such that lim(n-->infinity)M(-n) = 0. We are interested in the question, for which sequences a will there exist a solution to the functional equation with each function phi(j), j = 1,..., r, belonging to the Sobolev space W-p(k)(R-S)? Our approach will be to consider the convergence of the cascade algorithm. The cascade operator Q(a) associated with the sequence a is defined by QaF := Sigma(alphais an element ofZs) a(alpha)F(M-.-alpha). F is an element of (W-p(k)(R-s))(r). Let phi(0) be a nontrivial r x 1 vector of compactly supported functions in W-p(k)(R-S). The iteration scheme phi(n) = Q(a)phi(n-1), n = 1, 2, ..., is called a cascade algorithm, or a subdivision scheme. Under natural assumptions on a, a feasible set of initial vectors is identified from the conditions on an initial vector implied by the convergence of the subdivision scheme. These conditions are determined by the matrix A(0) = m(-1) Sigma(alphais an element ofZs) a(alpha), m = \det M\, and are related to polynomial reproducibility and the classical Strang-Fix conditions. The formal definition of convergence in the Sobolev norm for the subdivision scheme is that the scheme will converge for any choice of initial vector from the feasible set (to the same solution phi). We give a characterization for this concept of convergence in terms of the p-norm joint spectral radius of a finite collection of transition operators determined by the sequence a restricted to a certain invariant subspace. The invariant subspace is intimately connected to the Strang-Fix type conditions that determine the feasible set of initial vectors. (C) 2002 Elsev

关键词： refinement equations multiple refinable functions vector subdivision schemes cascade algorithm joint spectral radii transition operators

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multiple refinable Hermite interpolants

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JOURNAL OF APPROXIMATION THEORY 2000年第1期102卷 46-71页

作者： Zhou, DX City Univ Hong Kong Dept Math Hong Kong Peoples R China

where phi = (phi(1),..., phi(r))(T) is a vector of compactly supported functions on R and a is a finitely supported sequence of r x r matrices called the refinement mask. If phi is a continuous solution and a is supported on [N-1, N-2], then v := (phi(n))(n=N1)(N2-1) is an eigenvector of the matrix (a(2k - n))(k, n = N1)(N2 - 1) associated with eigenvalue 1. Conversely, given such an eigenvector v, we may ask whether there exists a continuous solution phi such that phi(n) = v(n) for N-1 less than or equal to n less than or equal to N-2 - 1 (phi(n) = 0 for n is not an element of [N-1 . N2 - 1]. according to the support ). The first part of this paper answers this question completely. This existence problem is more general than either the convergence of the subdivision scheme or the requirement of stability, since in one of the latter cases, the eigenvector v, is unique up to a constant multiplication. The second part of this paper is concerned with Hermite interpolant solutions, i.e., fur some n(0) epsilon Z and j m = 1,..., r, phi(f) epsilon Cr-1(R) and phi(f)((m-1)) (n) = delta(j, m)delta(n, n0), n epsilon Z. We provide a necessary and sufficient condition for the refinement equation to have an Hermite interpolant solution. The condition is strictly in terms of the refinement mask. Our method is to characterize the existence and the Hermite interpolant condition by joint spectral radii of matrices. Several concrete examples are presented to illustrate the general theory. (C) 2000 Academic Press.

关键词： refinement equations Hermite interpolants multiple refinable functions joint spectral radii

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Convergence rates of vector cascade algorithms in L_p

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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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Vector subdivision schemes and multiple wavelets

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MATHEMATICS OF COMPUTATION 1998年第224期67卷 1533-1563页

作者： Jia, RQ Riemenschneider, SD Zhou, DX Univ Alberta Dept Math Sci Edmonton AB T6G 2G1 Canada City Univ Hong Kong Dept Math Kowloon Peoples R China

We consider solutions of a system of refinement equations written in the form phi = Sigma(alpha is an element of Z) a(alpha)phi(2 . - alpha), where the vector of functions phi = (phi(1),..., phi(r))(T) is in (L-p(R))(r) and a is a finitely supported sequence of r x r matrices called the refinement mask. Associated with the mask a is a linear operator Q(a) defined on (L-p(R))(r) by Q(af) := Sigma(alpha is an element of Z) a(alpha) f(2 . - alpha). This paper is concerned with the convergence of the subdivision scheme associated with a, i.e., the convergence of the sequence (Q(a)(n) f)(n=1,2,...) in the L-p-norm. Our main result characterizes the convergence of a subdivision scheme associated with the mask a in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the L(2-)convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations. Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.

关键词： refinement equations multiple refinable functions multiple wavelets vector subdivision schemes joint spectral radii transition operators

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