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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Multivariate matrix refinable functions
with arbitrary matrix dilation


Author: Qingtang Jiang
Journal: Trans. Amer. Math. Soc. 351 (1999), 2407-2438
MSC (1991): Primary 39B62, 42B05, 41A15; Secondary 42C15
Published electronically: February 15, 1999
MathSciNet review: 1650101
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Abstract: Characterizations of the stability and orthonormality of a multivariate matrix refinable function $\Phi$ 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 $\Phi$ is equivalent to the order of the vanishing moment conditions of the matrix refinement mask $\{\mathbf{P}_{\alpha}\}$. The restricted transition operator associated with the matrix refinement mask $\{\mathbf{P}_{\alpha}\}$ is represented by a finite matrix $({\mathcal A} _{Mi-j})_{i, j}$, with ${\mathcal A} _j=|\hbox{det$(M)$}|^{-1}\sum _{\kappa }\mathbf{P}_{\kappa -j}\otimes \mathbf{P}_{\kappa }$ and $\mathbf{P}_{\kappa -j}\otimes \mathbf{P}_{\kappa }$ being the Kronecker product of matrices $\mathbf{P}_{\kappa -j}$ and $\mathbf{P}_{\kappa }$. The spectral properties of the transition operator are studied. The Sobolev regularity estimate of a matrix refinable function $\Phi$ 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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Additional Information

Qingtang Jiang
Affiliation: Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 and Department of Mathematics, Peking University, Beijing 100871, China
Email: qjiang@haar.math.nus.edu.sg

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02449-6
PII: S 0002-9947(99)02449-6
Keywords: Matrix refinable function, transition operator, stability, orthonormality, approximation order, regularity
Received by editor(s): September 26, 1996
Published electronically: February 15, 1999
Additional Notes: The author was supported by an NSTB post-doctoral research fellowship at the National University of Singapore.
Article copyright: © Copyright 1999 American Mathematical Society