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In this paper we give some criteria for the existence of compactly supported $C^{k+\alpha }$-solutions ($k$ is an integer and $0\le \alpha <1$) of matrix refinement equations. Several examples are presented to illustrate the general theory.

Let A be a d × d integral expanding matrix and let $S_j\left(x\right)=A^{-1}(x+d_j)$ for some $d_j\in {\mathbb{Z}}^d$, j = 1,...,m. The iterated function system (IFS) ${S_j}_{j=1}^m$ generates self-affine measures and scale functions. In general this IFS has overlaps, and it is well known that in many special cases the analysis of such measures or functions is facilitated by expressing them in vector-valued forms with respect to another IFS that satisfies the open set condition. In this paper we prove a general theorem on such representation. The proof is constructive;...