Memoirs of the American Mathematical Society 2004; 82 pp; softcover Volume: 170 ISBN10: 0821835203 ISBN13: 9780821835203 List Price: US$60 Individual Members: US$36 Institutional Members: US$48 Order Code: MEMO/170/807
 Let \(A\) be a dilation matrix, an\(n \times n\) expansive matrix that maps a fullrank 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 \(f(x) = \sum_{k \in \Lambda} c_k \, f(Axk)\). 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 \(f^{(i+1)}(x) = \sum_{k \in \Lambda} c_k \, f^{(i)}(Axk)\), 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  Introduction
 Matrices, tiles, and the joint spectral radius
 Generalized selfsimilarity and the refinement equation
 Multiresolution analysis
 Examples
 Bibliography
 Appendix A. Index of symbols
