Frame wavelets in subspaces of L2(Rd)

Let A be a d x d real expansive matrix. We characterize the reducing subspaces of L-2(R-d) for A-dilation and the regular translation operators acting on L-2(R-d). We also characterize the Lebesgue measurable subsets E of R d such that the function defined by inverse Fourier transform of [1/(2pi)(d/2)]chi(E) generates through the same A-dilation and the regular translation operators a normalized tight frame for a given reducing subspace. We prove that in each reducing subspace, the set of all such functions is nonempty and is also path connected in the regular L-2 (R-d)-norm.