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

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Graduate students and research mathematicians interested in applied mathematics.