Fredholm properties of multivariate refinement equations

The solvability of multivariate refinement equations in the Hilbert space L2(Rs) is studied. It is shown that the set of all L2-solutions of such equations either consists of the trivial solution only or it contains a subspace isomorphic to a space L(infinity)(SM) where SM is a subset of Rs with a positive Lebesgue measure. Therefore, the corresponding multivariate refinement operator is Fredholm if and only if it is invertible.