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Self-Similarity and Multiwavelets in Higher Dimensions
Carlos A. Cabrelli, University of Buenos Aires, Argentina, Christopher Heil, Georgia Institute of Technology, Atlanta, GA, and Ursula M. Molter, University of Buenos Aires, Argentina
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Memoirs of the American Mathematical Society
2004; 82 pp; softcover
Volume: 170
ISBN-10: 0-8218-3520-3
ISBN-13: 978-0-8218-3520-3
List Price: US$57 Individual Members: US$34.20
Institutional Members: US\$45.60
Order Code: MEMO/170/807

Let $$A$$ be a dilation matrix, an$$n \times n$$ expansive matrix that maps a full-rank lattice $$\Gamma \subset \mathbf{R}^n$$ into itself. Let $$\Lambda$$ be a finite subset of $$\Gamma$$, and for $$k \in \Lambda$$ let $$c_k$$ be $$r \times r$$ complex matrices. The refinement equation corresponding to $$A$$, $$\Gamma$$, $$\Lambda$$, and $$c = \{c_k\}_{k \in \Lambda}$$ is $$f(x) = \sum_{k \in \Lambda} c_k \, f(Ax-k)$$. A solution $$f \,\colon\, \mathbf{R}^n \to \mathbf{C}^r$$, if one exists, is called a refinable vector function or a vector scaling function of multiplicity $$r$$. In this manuscript we characterize the existence of compactly supported $$L^p$$ or continuous solutions of the refinement equation, in terms of the $$p$$-norm joint spectral radius of a finite set of finite matrices determined by the coefficients $$c_k$$. We obtain sufficient conditions for the $$L^p$$ convergence ($$1 \le p \le \infty$$) of the Cascade Algorithm $$f^{(i+1)}(x) = \sum_{k \in \Lambda} c_k \, f^{(i)}(Ax-k)$$, and necessary conditions for the uniform convergence of the Cascade Algorithm to a continuous solution. We also characterize those compactly supported vector scaling functions which give rise to a multiresolution analysis for $$L^2(\mathbf{R}^n)$$ of multiplicity $$r$$, and provide conditions under which there exist corresponding multiwavelets whose dilations and translations form an orthonormal basis for $$L^2(\mathbf{R}^n)$$.