On Existence and Weak Stability of Matrix Refinable Functions

Jiang, Qingtang; Shen, Zuowei

doi:10.1007/s003659900111

Cited by 53 publications

(46 citation statements)

References 4 publications

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“…About the existence of L 2 -solutions of (1.1) for M = 2I r , a similar result was obtained in [21]. For the special case r = 1 and d = 1, this result was given in [28].…”

Section: Lemma 29 For Any Hsupporting

confidence: 71%

“…By Proposition 2.4, the following proposition can be obtained as in [21]. Its proof is presented here for the sake of completeness.…”

Section: Stability and Orthonormalitymentioning

confidence: 88%

“…(2.13) Equation (2.13) can be obtained as in [21] for the case M = 2I r , and we omit the details here.…”

Section: Lemma 29 For Any Hmentioning

confidence: 99%

“…For M = 2I r , r ≥ 1, where I r is the r × r identity matrix, the characterizations of the stability and orthonormality of a matrix refinable function Φ were provided in terms of the mask in [26]; the regularity estimates of Φ were studied in [26], [19], and in [3], [24] for the case d = 1; the existence of the distribution solution of (1.1) and the characterization of the weak stability of solutions of (1.1) were discussed in [21]. In the construction of multivariate wavelets, the dilation matrix M is involved.…”

Section: Introductionmentioning

confidence: 99%

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Multivariate matrix refinable functions with arbitrary matrix dilation

Jiang

1999

Trans. Amer. Math. Soc.

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Abstract. Characterizations of the stability and orthonormality of a multivariate matrix refinable function Φ with arbitrary matrix dilation M are provided in terms of the eigenvalue and 1-eigenvector properties of the restricted transition operator. Under mild conditions, it is shown that the approximation order of Φ is equivalent to the order of the vanishing moment conditions of the matrix refinement mask {Pα}. The restricted transition operator associated with the matrix refinement mask {Pα} is represented by a finite matrix (A Mi−j ) i,j , with A j = |det(M )| −1 κ P κ−j ⊗ Pκ and P κ−j ⊗ Pκ being the Kronecker product of matrices P κ−j and Pκ. The spectral properties of the transition operator are studied. The Sobolev regularity estimate of a matrix refinable function Φ is given in terms of the spectral radius of the restricted transition operator to an invariant subspace. This estimate is analyzed in an example.

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“…About the existence of L 2 -solutions of (1.1) for M = 2I r , a similar result was obtained in [21]. For the special case r = 1 and d = 1, this result was given in [28].…”

Section: Lemma 29 For Any Hsupporting

confidence: 71%

“…By Proposition 2.4, the following proposition can be obtained as in [21]. Its proof is presented here for the sake of completeness.…”

Section: Stability and Orthonormalitymentioning

confidence: 88%

“…(2.13) Equation (2.13) can be obtained as in [21] for the case M = 2I r , and we omit the details here.…”

Section: Lemma 29 For Any Hmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multivariate matrix refinable functions with arbitrary matrix dilation

Jiang

1999

Trans. Amer. Math. Soc.

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“…For vector homogeneous refinement equations, see [9], [2], [22], [19], and [27]. Nonhomogeneous refinement equations generalized from their homogeneous counterpart are motivated by constructions of multiwavelets to obtain multi-channel filters with good time-frequency localization in the area of signal processing and constructions of wavelets on a finite interval to find numerical solutions of differential equations in the area of numerical analysis (see, e.g., [24] and [3]).…”

Section: φ R )mentioning

confidence: 99%

Convergence of cascade algorithms associated with nonhomogeneous refinement equations

Jiang

Jia

2000

Proc. Amer. Math. Soc.

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Abstract. This paper is devoted to a study of multivariate nonhomogeneous refinement equations of the form. . , gr) T is a given vector of functions on R s , M is an s × s dilation matrix, and a is a finitely supported refinement mask such that each a(α) is an r × r (complex) matrix. Let φ 0 be an initial vector in (L 2 (R s )) r . The corresponding cascade algorithm is given byIn this paper we give a complete characterization for the L 2 -convergence of the cascade algorithm in terms of the refinement mask a, the nonhomogeneous term g, and the initial vector of functions φ 0 .

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Symmetric–Antisymmetric Orthonormal Multiwavelets and Related Scalar Wavelets

Shen

Tan

Tham

2000

Applied and Computational Harmonic Analysis

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For compactly supported symmetric-antisymmetric orthonormal multiwavelet systems with multiplicity 2, we first show that any length-2N multiwavelet system can be constructed from a length-(2N + 1) multiwavelet system and vice versa. Then we present two explicit formulations for the construction of multiwavelet functions directly from their associated multiscaling functions. This is followed by the relationship between these multiscaling functions and the scaling functions of related orthonormal scalar wavelets. Finally, we present two methods for constructing families of symmetric-antisymmetric orthonormal multiwavelet systems via the construction of the related scalar wavelets.

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On Existence and Weak Stability of Matrix Refinable Functions

Cited by 53 publications

References 4 publications

Multivariate matrix refinable functions with arbitrary matrix dilation

Multivariate matrix refinable functions with arbitrary matrix dilation

Convergence of cascade algorithms associated with nonhomogeneous refinement equations

Symmetric–Antisymmetric Orthonormal Multiwavelets and Related Scalar Wavelets

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