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Stability implies convergence of cascade algorithms in Sobolev space

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2002年第1期268卷 41-52页

作者： Chen, DR Zheng, XB Beijing Univ Aeronaut & Astronaut Dept Appl Math Beijing 100083 Peoples R China Hubei Inst Nationalities Dept Math Enshi 445000 Hubei Peoples R China

This article proves that the stability of the shifts of a refinable function vector ensures the convergence of the corresponding cascade algorithm in Sobolev space to which the refinable function vector belongs. An ex... 详细信息

关键词： cascade algorithm Sobolev space stability Hermite interpolant

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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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Norms concerning subdivision sequences and their applications in wavelets

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APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 2001年第3期11卷 329-346页

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

Let a := (a(alpha))(alpha is an element ofZ)s be a finitely supported sequence of r x r matrices and M be a dilation matrix. The subdivision sequence {(a(n)(alpha))(alpha is an element ofZ)s : n is an element of N} is defined by a(1) = a and [GRAPHICS] Let 1 less than or equal to p less than or equal to infinity and f = (f(l),...,f(r))(T) be a vector of compactly supported functions in L-p (RI). The stability is not assumed for f. The purpose of this paper is to give a formula for the asymptotic behavior of the L-p-norms of the combinations of the shifts of f with the subdivision sequence coefficients: [GRAPHICS] Such an asymptotic behavior plays an essential role in the investigation of wavelets and subdivision schemes. In this paper we show some applications in the convergence of cascade algorithms, construction of inhomogeneous multiresolution analyzes, and smoothness analysis of refinable functions. Some examples are provided to illustrate the method. (C) 2001 Academic Press.

关键词： subdivision sequence joint spectral radius cascade algorithm inhomogeneous refinement equation wavelets smoothness

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Subdivision schemes and refinement equations with nonnegative masks

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JOURNAL OF APPROXIMATION THEORY 2001年第2期113卷 207-220页

作者： Wang, Y Georgia Inst Technol Sch Math Atlanta GA 30332 USA

We consider the two-scale refinement equation f(x) = Sigma(n=0)(N) c(n)f(2x - n) with Sigma(n) c(2n) = Sigma(n)c(2n+1) = 1 where c(0), c(N) not equal 0 and the corresponding subdivision scheme. We study the convergence of the subdivision scheme and the cascade algorithm when all c(n) greater than or equal to 0. It has long been conjectured that under such an assumption the subdivision algorithm converge, and the cascade algorithm converge uniformly to a continuous function, if and only if only if 0 < c(0), c(N) < 1 and the greatest common divisor of S = {n: c(n) > 0} is 1. We prove the conjecture for a large class of refinement equations. (C) 2001 Academic Press.

关键词： nonnegative mask cascade algorithm subdivision scheme refinement equation refutable function

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The stability of subdivision operator at its fixed point

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SIAM JOURNAL ON MATHEMATICAL ANALYSIS 2001年第2期33卷 448-460页

作者： Protasov, V Moscow MV Lomonosov State Univ Dept Mech & Math Moscow 119899 Russia

We consider the univariate two-scale refinement equation phi (x) = Sigma (N)(k=0) c(k)phi (2x - k), where c(0),..., c(N) are complex values and Sigmac(k) = 2. This paper analyzes the correlation between the existence of smooth compactly supported solutions of this equation and the convergence of the corresponding cascade algorithm/subdivision scheme. We introduce a criterion that expresses this correlation in terms of the mask of the equation. We show that the convergence of the subdivision scheme depends on values that the mask takes at the points of its generalized cycles. This means in particular that the stability of shifts of refinable function is not necessary for the convergence of the subdivision process. This also leads to some results on the degree of convergence of subdivision processes and on factorizations of refinable functions.

关键词： refinement equations cascade algorithm subdivision process degree of convergence stability cycles tree

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Nonuniform cascade algorithms

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2000年第1-2期119卷 223-234页

作者： Goodman, TNT Lee, SL Univ Dundee Dept Math Dundee DD1 4HN Scotland Natl Univ Singapore Dept Math Singapore 119260 Singapore

For a sequence of bounded linear operators H-k, k = 0, 1,..., on a Banach space S, the algorithm phi(k,n) = H(k)phi(k+1,n-1) generates a family of sequences (phi(k,n))(n=0)(infinity), k = 0, 1,..., from an initial family of vectors phi(k,0) epsilon S, k = 0, 1,.... We study the convergence of phi(k,n) as n --> infinity, and give an application on the convergence of cascade algorithms for nonuniform splines when S is the space of all sequences phi:=(phi(i))(i epsilon z) with norm parallel to phi parallel to:=sup(i epsilon z)parallel to phi(i)parallel to < infinity, and phi(i), i epsilon Z, belong to the Banach space X = L-2(R). (C) 2000 Elsevier Science B.V. All rights reserved.

关键词： Banach space cascade algorithm operator refinement equation

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A complete solution characterizing smooth refinable functions

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SIAM JOURNAL ON MATHEMATICAL ANALYSIS 2000年第6期31卷 1332-1350页

作者： Protasov, V Erasmus Univ Dept Econ NL-3062 PA Rotterdam Netherlands Inst Adv Study Princeton NJ 08540 USA

In this paper we study univariate two-scale refinement equations phi(x) = Sigma(k is an element of Z)c(k)phi (2x - k), where the coefficients c(k) is an element of C satisfy an exponential decay assumption. We show that any refinement equation that has a smooth solution can be reduced to the well-studied case of complete sum rules: Sigma(k)(-1)(k)k(n)c(k) = 0, n = 0,..., L, where L depends on regularity of the solution. This result makes it possible to extend previously known results on refinable functions and subdivision schemes from the case of complete sum rules to the general case. As a corollary we obtain sharp necessary conditions for the existence of smooth refinable functions and the convergence of corresponding cascade algorithms. Other applications concern polynomial spaces spanned by integer translates of a refinable function and one special property of linear operators associated to refinement equations.

关键词： refinement equations regularity reducibility tree cascade algorithm

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Spline interpolation and wavelet construction

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APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 1998年第3期5卷 249-276页

作者： Lee, SL Sharma, A Tan, HH Natl Univ Singapore Dept Math Singapore 119260 Singapore Univ Alberta Dept Math Edmonton AB T6G 2G1 Canada

The method of Dubuc and Deslauriers on symmetric interpolatory subdivision is extended to study the relationship between interpolation processes and wavelet construction. Refinable and interpolatory functions are constructed in stages from B-splines. Their method constructs the filter sequence (its Laurent polynomial) of the interpolatory function as a product of Laurent polynomials. This provides a natural way of splitting the filter for the construction of orthonormal and biorthogonal scaling functions leading to orthonormal and biorthogonal wavelets. Their method also leads to a class of filters which includes the minimal length Daubechies compactly supported orthonormal wavelet coefficients. Examples of ''good'' filters are given together with results of numerical experiments conducted to test the performance of these filters in data compression. (C) 1998 Academic Press.

关键词： refinable function wavelets interpolatory function uniform B-spline Euler-Frobenius polynomial Riesz basis biorthogonal basis orthonormal basis transition operator subdivision algorithm cascade algorithm condition E

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Refinable function vectors

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SIAM JOURNAL ON MATHEMATICAL ANALYSIS 1998年第1期29卷 235-250页

作者： Shen, ZW Natl Univ Singapore Dept Math Singapore 119260 Singapore

Refinable function vectors are usually given in the form of an infinite product of their refinement (matrix) masks in the frequency domain and approximated by a cascade algorithm in both time and frequency domains. We provide necessary and sufficient conditions for the convergence of the cascade algorithm. We also give necessary and sufficient conditions for the stability and orthonormality of refinable function vectors in terms of their refinement matrix masks. Regularity of function vectors gives smoothness orders in the time domain and decay rates at infinity in the frequency domain. Regularity criteria are established in terms of the vanishing moment order of the matrix mask.

关键词： refinable function vectors stable basis cascade algorithm regularity

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Inhomogeneous refinement equations

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 1998年第6期4卷 733-747页

作者： Strang, G Zhou, DX MIT Dept Math Cambridge MA 02139 USA City Univ Hong Kong Dept Math Kowloon Hong Kong

Equations with two time scales (refinement equations or dilation equations) are central to wavelet theory. Several applications also include an inhomogeneous forcing term F(t). We develop here a parr of the existence theory for the inhomogeneous refinement equation [GRAPHICS] where a(k) is a finite sequence and F is a compactly supported distribution on R. The existence of compactly supported distributional solutions to an inhomogeneous refinement equation is characterized in terms of conditions on the pair (a, F). To have L-p solutions from F is an element of L-p(R), we construct by the cascade algorithm a sequence of functions {phi(n)} from a compactly supported initial function phi(o) is an element of L-p (R) as [GRAPHICS] A necessary and sufficient condition for the sequence {phi(n)} to converge in L-p(R)(1 less than or equal to p less than or equal to infinity) is given by the p-norm joint spectral radius of two matrices derived from the mask a. A convexity property of the p-norm joint spectral radius (1 less than or equal to p less than or equal to infinity) is presented. Finally, the general theory is applied to some examples and multiple refinable functions.

关键词： inhomogeneous refinement equation wavelets joint spectral radius cascade algorithm multiple refinable function

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