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

DOI:
https://doi.org/10.1090/S0002-9947-99-02449-6

Published electronically:
February 15, 1999

MathSciNet review:
1650101

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Abstract | References | Similar Articles | Additional Information

Abstract: Characterizations of the stability and orthonormality of a multivariate matrix refinable function with arbitrary matrix dilation are provided in terms of the eigenvalue and -eigenvector properties of the restricted transition operator. Under mild conditions, it is shown that the approximation order of is equivalent to the order of the vanishing moment conditions of the matrix refinement mask . The restricted transition operator associated with the matrix refinement mask is represented by a finite matrix , with and being the Kronecker product of matrices and . The spectral properties of the transition operator are studied. The Sobolev regularity estimate of a matrix refinable function 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:
https://doi.org/10.1090/S0002-9947-99-02449-6

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