# Self-Similarity and Multiwavelets in Higher Dimensions

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*Carlos A. Cabrelli; Christopher Heil; Ursula M. Molter*

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

#### Table of Contents

# Table of Contents

## Self-Similarity and Multiwavelets in Higher Dimensions

- Contents v6 free
- Acknowledgments vii8 free
- Chapter 1. Introduction 110 free
- Chapter 2. Matrices, Tiles and the Joint Spectral Radius 514
- Chapter 3. Generalized Self–Similarity and the Refinement Equation 1928
- 3.1. Generalized Self–Similarity 1928
- 3.2. Sufficient Conditions for the Existence of Vector Scaling Functions 2029
- 3.3. Continuous Solutions and the Support of the Refinement Equation Coefficients 2635
- 3.4. Higher–Order Accuracy 2736
- 3.5. Invariant Subspaces 3241
- 3.6. Necessary Conditions for the Existence of Continuous Vector Scaling Functions 3645
- 3.7. Holder Continuity 4655

- Chapter 4. Multiresolution Analysis 4958
- Chapter 5. Examples 6372
- Bibliography 7786
- Appendix A. Index of Symbols 8190