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Title: Multivariate matrix refinable functions with arbitrary matrix dilation
Authors: Jiang, Q. 
Keywords: Approximation order
Matrix refinable function
Transition operator
Issue Date: 1999
Citation: Jiang, Q. (1999). Multivariate matrix refinable functions with arbitrary matrix dilation. Transactions of the American Mathematical Society 351 (6) : 2407-2438. ScholarBank@NUS Repository.
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 (AMi-j)i,j, with Aj = |det(M)|-1 ∑κ Pκ-j ⊗ Pκ and Pκ-j ⊗ Pκ being the Kronecker product of matrices Pκ-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. ©1999 American Mathematical Society.
Source Title: Transactions of the American Mathematical Society
ISSN: 00029947
Appears in Collections:Staff Publications

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