Vector subdivision schemes and multiple wavelets

Authors:
Rong-Qing Jia, S. D. Riemenschneider and Ding-Xuan Zhou

Journal:
Math. Comp. **67** (1998), 1533-1563

MSC (1991):
Primary 39B12, 41A25, 42C15, 65F15

DOI:
https://doi.org/10.1090/S0025-5718-98-00985-5

MathSciNet review:
1484900

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider solutions of a system of refinement equations written in the form \begin{equation*}\phi = \sum _{\alpha \in \mathbb {Z}} a(\alpha )\phi (2\cdot -\alpha ),\end{equation*} where the vector of functions $\phi =(\phi ^{1},\ldots ,\phi ^{r})^{T}$ is in $(L_{p}(\mathbb {R}))^{r}$ and $a$ is a finitely supported sequence of $r\times r$ matrices called the refinement mask. Associated with the mask $a$ is a linear operator $Q_{a}$ defined on $(L_{p}(\mathbb {R}))^{r}$ by $Q_{a} f := \sum _{\alpha \in \mathbb {Z}} a(\alpha )f(2\cdot -\alpha )$. This paper is concerned with the convergence of the subdivision scheme associated with $a$, i.e., the convergence of the sequence $(Q_{a}^{n}f)_{n=1,2,\ldots }$ in the $L_{p}$-norm. Our main result characterizes the convergence of a subdivision scheme associated with the mask $a$ in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the $L_{2}$-convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations. Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.

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Additional Information

**Rong-Qing Jia**

Email:
jia@xihu.math.ualberta.ca

**S. D. Riemenschneider**

Affiliation:
Department of Mathematical Sciences, University of Alberta, Edmonton, Canada T6G 2G1

Email:
sherm@approx.math.ualberta.ca

**Ding-Xuan Zhou**

Affiliation:
Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

Email:
mazhou@math.cityu.edu.hk

Keywords:
Refinement equations,
multiple refinable functions,
multiple wavelets,
vector subdivision schemes,
joint spectral radii,
transition operators

Received by editor(s):
December 12, 1996

Additional Notes:
Research supported in part by NSERC Canada under Grants # OGP 121336 and A7687.

Article copyright:
© Copyright 1998
American Mathematical Society