Supports of Locally Linearly Independent M-Refinable Functions, Attractors of Iterated Function Systems and Tilings

We study supports of M -re ̄nable functions by means of attractors of iterated function systems, where M is an integer greater than (or equal to) 2. Most of the time, the re ̄nable function Á is not explicitly known. We need to study their properties from the re ̄nement mask. Under the assumption of local linear independence, we show that the support equals to the attractor of an iterated function system. So the local linear independence of shifts of M -re ̄nable functions is required. A complete characterization for this local linear independence property is given by ̄nite matrix products, strictly in terms of the re ̄nement mask. We do this in a more general setting, the vector re ̄nement equations. A connection between self-a±ne tilings and L2 solutions of re ̄nement equations without assuming the basic sum rule is pointed out, which leads to further problems. Several examples are provided to illustrate the general theory.