The existence of subspace wavelet sets

Let H be a reducing subspace of L-2(R-d) that is, a closed subspace of L-2(R-d) with the property that f(A(m)t - l) is an element of H for any f is an element of H, m is an element of Z and l is an element of Z(d), where A is a d x d expansive matrix. It is known that H is a reducing subspace if and only if there exists a measurable subset M of R-d such that A(t)M = M and F(H) = L-2(R-d) (.) chi(M). Under some given conditions of M, it is known that there exist A-dilation subspace wavelet sets with respect to H. In this paper, we prove that this holds in general. (C) 2003 Elsevier Science B.V. All rights reserved.