Frame wavelets in subspaces of $L^2(\mathbb R^d)$

Let $A$ be a $d\times d$ real expansive matrix. We characterize the reducing subspaces of $L^2(\mathbb{R}^d)$for $A$-dilation and the regular translation operators acting on $L^2 (\mathbb{R}^d).$We also characterize the Lebesgue measurable subsets $E$of $\mathbb{R}^d$ such that the function defined by inverse Fourier transform of $[1/(2\pi)^{d/2}]\chi_{E}$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 $L^2(\mathbb{R}^d)$-norm.