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Self-Similarity and Multiwavelets in Higher Dimensions

Carlos A. Cabrelli University of Buenos Aires, Buenos Aires, Argentina
Christopher Heil Georgia Institute of Technology, Atlanta, GA
Ursula M. Molter University of Buenos Aires, Buenos Aires, Argentina
Available Formats:
Electronic ISBN: 978-1-4704-0408-6
Product Code: MEMO/170/807.E
List Price: $60.00 MAA Member Price:$54.00
AMS Member Price: $36.00 Click above image for expanded view Self-Similarity and Multiwavelets in Higher Dimensions Carlos A. Cabrelli University of Buenos Aires, Buenos Aires, Argentina Christopher Heil Georgia Institute of Technology, Atlanta, GA Ursula M. Molter University of Buenos Aires, Buenos Aires, Argentina Available Formats:  Electronic ISBN: 978-1-4704-0408-6 Product Code: MEMO/170/807.E  List Price:$60.00 MAA Member Price: $54.00 AMS Member Price:$36.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 1702004; 82 pp
MSC: Primary 39;

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 $L^2(\mathbf{R}^n)$.

Graduate students and research mathematicians interested in applied mathematics.

• Chapters
• 1. Introduction
• 2. Matrices, tiles and the joint spectral radius
• 3. Generalized self-similarity and the refinement equation
• 4. Multiresolution analysis
• 5. Examples
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Volume: 1702004; 82 pp
MSC: Primary 39;

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 $L^2(\mathbf{R}^n)$.