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72
Vector cascade algorithms and refinable function vectors in Sobolev spaces
 J. Approx. Theory
, 2002
"... In this paper we shall study vector cascade algorithms and refinable function vectors with a general isotropic dilation matrix in Sobolev spaces. By investigating several properties of the initial function vectors in a vector cascade algorithm, we are able to take a relatively unified approach to st ..."
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Cited by 54 (35 self)
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In this paper we shall study vector cascade algorithms and refinable function vectors with a general isotropic dilation matrix in Sobolev spaces. By investigating several properties of the initial function vectors in a vector cascade algorithm, we are able to take a relatively unified approach to study several questions such as convergence, rate of convergence and error estimate for a perturbed mask of a vector cascade algorithm in a Sobolev space W k p (R s)(1 � p � ∞, k ∈ N∪{0}). We shall characterize the convergence of a vector cascade algorithm in a Sobolev space in various ways. As a consequence, a simple characterization for refinable Hermite interpolants and a sharp error estimate for a perturbed mask of a vector cascade algorithm in a Sobolev space will be presented. The approach in this paper enables us to answer some unsolved questions in the literature on vector cascade algorithms and to comprehensively generalize and improve results on scalar cascade algorithms and scalar refinable functions to the vector case. Key words: vector cascade algorithm, vector subdivision scheme, refinable function vector, Hermite interpolant, initial function vector, error estimate, sum rules, smoothness.
Spectral analysis of the transition operator and its applications to smoothness analysis of wavelets
 SIAM J. Matrix. Anal. Appl
, 2001
"... The purpose of this paper is to investigate spectral properties of the transition operator associated to a multivariate vector refinement equation and their applications to the study of smoothness of the corresponding refinable vector of functions. Let Φ = (φ1,..., φr) T be an r × 1 vector of compac ..."
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Cited by 43 (17 self)
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The purpose of this paper is to investigate spectral properties of the transition operator associated to a multivariate vector refinement equation and their applications to the study of smoothness of the corresponding refinable vector of functions. Let Φ = (φ1,..., φr) T be an r × 1 vector of compactly supported functions in L2(IR s) satisfying the refinement equation Φ = � α ∈ Zs a(α)Φ(M · − α), where M is an expansive integer matrix. We assume that M is isotropic, i.e., M is similar to a diagonal matrix diag(σ1,..., σs) with σ1  = · · · = σs. For µ = (µ1,..., µs) ∈ IN s 0, define. The smoothness of Φ is measured by the critical exponent σ −µ: = σ −µ1
Computing the Smoothness Exponent of a Symmetric Multivariate Refinable Function
, 2003
"... Smoothness and symmetry are two important properties of a refinable function. It is known that the Sobolev smoothness exponent of a refinable function can be estimated by computing the spectral radius of certain finite matrix which is generated from a mask. However, the increase of dimension and the ..."
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Cited by 43 (26 self)
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Smoothness and symmetry are two important properties of a refinable function. It is known that the Sobolev smoothness exponent of a refinable function can be estimated by computing the spectral radius of certain finite matrix which is generated from a mask. However, the increase of dimension and the support of a mask tremendously increases the size of the matrix and therefore make the computation very expensive. In this paper, we shall present a simple algorithm to efficiently numerically compute the smoothness exponent of a symmetric refinable function with a general dilation matrix. By taking into account of symmetry of a refinable function, our algorithm greatly reduces the size of the matrix and enables us to numerically compute the Sobolev smoothness exponents of a large class of symmetric refinable functions. Step by step numerically stable algorithms and details of the numerical implementation are given. To illustrate our results by performing some numerical experiments, we construct a family of dyadic interpolatory masks in any dimension and we compute the smoothness exponents of their refinable functions in dimension three. Several examples will also be presented for computing smoothness exponents of symmetric refinable functions on the quincunx lattice and on the hexagonal lattice.
Analysis And Construction Of Optimal Multivariate Biorthogonal Wavelets With Compact Support
 SIAM J. Math. Anal
, 1998
"... . In applications, it is well known that high smoothness, small support and high vanishing moments are the three most important properties of a biorthogonal wavelet. In this paper, we shall investigate the mutual relations among these three properties. A characterization of Lp (1 p 1) smoothness ..."
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Cited by 43 (33 self)
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. In applications, it is well known that high smoothness, small support and high vanishing moments are the three most important properties of a biorthogonal wavelet. In this paper, we shall investigate the mutual relations among these three properties. A characterization of Lp (1 p 1) smoothness of multivariate refinable functions is presented. It is well known that there is a close relation between a fundamental refinable function and a biorthogonal wavelet. We shall demonstrate that any fundamental refinable function, whose mask is supported on [1 \Gamma 2r; 2r \Gamma 1] s for some positive integer r and satisfies the sum rules of optimal order 2r, has Lp smoothness not exceeding that of the univariate fundamental refinable function with the mask br . Here the sequence br on Z is the unique univariate interpolatory refinement mask which is supported on [1 \Gamma 2r; 2r \Gamma 1] and satisfies the sum rules of order 2r. Based on a similar idea, we shall prove that any orthogonal...
Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix
 J. Comput. Appl. Math
"... Tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas. In this paper, for any d × d dilation matrix M, we demonstrate in a constructive way that we can construct compactly supported tight Mwavelet frames and orthonormal Mwavelet bases in ..."
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Cited by 33 (21 self)
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Tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas. In this paper, for any d × d dilation matrix M, we demonstrate in a constructive way that we can construct compactly supported tight Mwavelet frames and orthonormal Mwavelet bases in L2(R d) of exponential decay, which are derived from compactly supported Mrefinable functions, such that they can have both arbitrarily high smoothness and any preassigned order of vanishing moments. This paper improves several
Smoothness of multiple refinable functions and multiple wavelets
 SIAM J. Matrix Anal. Appl
, 1999
"... Abstract. We consider the smoothness of solutions of a system of refinement equations written in the form φ = a(α)φ(2 ·−α), α∈Z where the vector of functions φ =(φ1,...,φr) T is in (Lp(R)) r and a is a finitely supported sequence of r × r matrices called the refinement mask. We use the generalized L ..."
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Cited by 32 (8 self)
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Abstract. We consider the smoothness of solutions of a system of refinement equations written in the form φ = a(α)φ(2 ·−α), α∈Z where the vector of functions φ =(φ1,...,φr) T is in (Lp(R)) r and a is a finitely supported sequence of r × r matrices called the refinement mask. We use the generalized Lipschitz space Lip ∗ (ν, Lp(R)), ν>0, to measure smoothness of a given function. Our method is to relate the optimal smoothness, νp(φ), to the pnorm joint spectral radius of the block matrices Aε, ε =0,1, given by Aε =(a(ε+2α−β))α,β, when restricted to a certain finite dimensional common invariant subspace V. Denoting the pnorm joint spectral radius by ρp(A0V,A1V), we show that νp(φ) ≥ 1/p − log 2 ρp(A0V,A1V) with equality when the shifts of φ1,...,φr 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.
Optimal Interpolatory Subdivision Schemes in Multidimensional Spaces
 SIAM J. Numer. Anal
, 1997
"... . We analyse the approximation and smoothness properties of fundamental and refinable functions that arise from interpolatory subdivision schemes in multidimensional spaces. In particular, we provide a general way for the construction of bivariate interpolatory refinement masks such that the corresp ..."
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Cited by 26 (20 self)
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. We analyse the approximation and smoothness properties of fundamental and refinable functions that arise from interpolatory subdivision schemes in multidimensional spaces. In particular, we provide a general way for the construction of bivariate interpolatory refinement masks such that the corresponding fundamental and refinable functions attain the optimal approximation order and smoothness order. In addition, these interpolatory refinement masks are minimally supported and enjoy full symmetry. Several examples are explicitly computed. Key words. Interpolatory subdivision schemes, fundamental functions, refinement equations, Lagrange interpolation, approximation order, sum rules, smoothness AMS subject classifications. 65 D 05, 65 D 17, 41 A 25, 41 A 63, 42 C 15 x1. Introduction In this paper we are interested in fundamental and refinable functions with compact support. A function OE is said to be fundamental if OE is continuous, OE(0) = 1, and OE(ff) = 0 for all ff 2 Z s nf0g...
Dual wavelet frames and Riesz bases in Sobolev spaces
, 2007
"... Abstract. This paper generalizes the mixed extension principle in L2(R d) of [50] to a pair of dual Sobolev spaces H s (R d) and H −s (R d). In terms of masks for φ, ψ 1,..., ψ L ∈ H s (R d) and ˜φ, ˜ ψ 1,..., ˜ ψ L ∈ H −s (R d), simple sufficient conditions are given to ensure that (X s (φ; ψ 1,. ..."
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Cited by 26 (15 self)
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Abstract. This paper generalizes the mixed extension principle in L2(R d) of [50] to a pair of dual Sobolev spaces H s (R d) and H −s (R d). In terms of masks for φ, ψ 1,..., ψ L ∈ H s (R d) and ˜φ, ˜ ψ 1,..., ˜ ψ L ∈ H −s (R d), simple sufficient conditions are given to ensure that (X s (φ; ψ 1,..., ψ L), X −s ( ˜ φ; ˜ ψ 1,..., ˜ ψ L)) forms a pair of dual wavelet frames in (H s (R d), H −s (R d)), where X s (φ; ψ 1,..., ψ L): = {φ( · − k) : k ∈ Z d} ∪ � 2 j(d/2−s) ψ ℓ (2 j · −k) : j ∈ N0, k ∈ Z d, ℓ = 1,..., L �. For s> 0, the key of this general mixed extension principle is the regularity of φ, ψ 1,..., ψ L, and the vanishing moments of ˜ ψ 1,..., ˜ ψ L, while allowing ˜ φ, ˜ ψ 1,..., ˜ ψ L to be tempered distributions not in L2(R d) and ψ 1,..., ψ L to have no vanishing moments. So, the systems X s (φ; ψ 1,..., ψ L) and X −s ( ˜ φ; ˜ ψ 1,..., ˜ ψ L) may not be able to be normalized into a frame of L2(R d). As an example, we show that {2 j(1/2−s) Bm(2 j · −k) : j ∈ N0, k ∈ Z} is a wavelet frame in H s (R) for any 0 < s < m − 1/2, where Bm is the Bspline of order m. This simple construction is also applied to multivariate box splines to obtain wavelet frames with short supports, noting that it is hard to construct nonseparable multivariate wavelet frames with small supports. Applying this general mixed extension principle, we obtain and characterize dual Riesz bases (X s (φ; ψ 1,..., ψ L), X −s ( ˜ φ; ˜ ψ 1,..., ˜ ψ L)) in Sobolev spaces (H s (R d), H −s (R d)). For example, all interpolatory wavelet systems in [25] generated by an interpolatory refinable function φ ∈ H s (R) with s> 1/2 are Riesz bases of the Sobolev space H s (R). This general mixed extension principle also naturally leads to a characterization of the Sobolev norm of a function in terms of weighted norm of its wavelet coefficient sequence (decomposition sequence) without requiring that dual wavelet frames should be in L2(R d), which is quite different to other approaches in the literature. 1.
Construction of Multivariate Biorthogonal Wavelets by CBC Algorithm
 Adv. Comput. Math
, 1998
"... In applications, it is well known that short support, high vanishing moments and reasonable smoothness are the three most important properties of a biorthogonal wavelet. Based on our previous work on analysis and construction of optimal fundamental refinable functions and optimal biorthogonal wavele ..."
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Cited by 23 (11 self)
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In applications, it is well known that short support, high vanishing moments and reasonable smoothness are the three most important properties of a biorthogonal wavelet. Based on our previous work on analysis and construction of optimal fundamental refinable functions and optimal biorthogonal wavelets, in this paper, we shall discuss the mutual relations among these three properties. For example, we shall see that any orthogonal scaling function, which is supported on [0; 2r \Gamma 1] s for some positive integer r and has accuracy order r, has Lp (1 p 1) smoothness not exceeding that of the univariate Daubechies orthogonal scaling function which is supported on [0; 2r \Gamma 1]. Similar results hold true for fundamental refinable functions and biorthogonal wavelets. Then, we shall discuss the relation between symmetry and the smoothness of a refinable function. Next, we discuss the coset by coset (CBC) algorithm reported in Han [29] to construct biorthogonal wavelets with arbitrar...
Approximation power of refinable vectors of functions, in Wavelet analysis and applications
 AMS/IP Stud. Adv. Math
, 2002
"... In this paper we survey recent results on approximation power of refinable vectors of functions. Let Φ = (φ1,..., φr) T be an r × 1 vector of compactly supported functions in Lp(IR s) (1 ≤ p ≤ ∞). The first part of this paper is devoted to an investigation of approximation power of S(Φ), the shifti ..."
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Cited by 22 (13 self)
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In this paper we survey recent results on approximation power of refinable vectors of functions. Let Φ = (φ1,..., φr) T be an r × 1 vector of compactly supported functions in Lp(IR s) (1 ≤ p ≤ ∞). The first part of this paper is devoted to an investigation of approximation power of S(Φ), the shiftinvariant space generated from Φ. We review results on characterizations of the approximation order of S(Φ) and describe approximation schemes that achieve the optimal approximation order. We also give a selfcontained treatment of various equivalent forms of the StrangFix conditions. We say that Φ is refinable if Φ = � α ∈ Zs a(α)Φ(M · − α), where M is an expansive s × s integer matrix, and the refinement mask a is finitely supported. The second part of this paper is dedicated to a study of accuracy of Φ. We review results on characterizations of the accuracy of Φ in terms of the mask in both time and frequency domains. We also discuss the relationship between the accuracy of Φ and the sum rules associated with the mask. Examples are provided to illustrate the general theory.