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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
Self-Similarity and Multiwavelets in Higher Dimensions
eBook 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
Self-Similarity and Multiwavelets in Higher Dimensions
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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
eBook 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 \(L^2(\mathbf{R}^n)\).

    Readership

    Graduate students and research mathematicians interested in applied mathematics.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Matrices, tiles and the joint spectral radius
    • 3. Generalized self-similarity and the refinement equation
    • 4. Multiresolution analysis
    • 5. Examples
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
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 \(L^2(\mathbf{R}^n)\).

Readership

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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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