Generalized multi-resolution analyses and a construction procedure for all wavelet sets in \(\mathbb{R}^n\) (Q1973855)

Let \(A\) be a real integer \(n\times n\) matrix, for which all eigenvalues have absolute value greater than one (i.e., an expansive matrix) and define a unitary operator on \(L^2(R^n)\) by \([\delta_Af](x)= |\det A |^{1/2}f(Ax)\). A set \(W\subseteq R^n\) is a wavelet set (with respect to \(A\)) if the function \(\psi\) defined by \(\chi_W=\hat{\psi}\) is a wavelet (i.e., if \(\{|\det A|^{m/2} \psi(A^m \cdot -k)\}_{m\in Z, k\in Z^n}\) is an orthonormal basis for \(L^2(R^n)\)). A generalized multiresolution analysis (GMRA) associated to \(A\) is an increasing family of closed subspaces \(\{V_j\}\) of \(L^2(R)\) for which (i) \(V_{j+1}=\delta_A V_j\) for all \(j\), (ii) \(\cap V_j = \{0\}\) and \(\overline {\cup V_j}= L^2(R^n)\), (iii) \(V_0\) is invariant under the action of the translation operators \((\gamma_kf)(x)= f(x+k), k\in Z^n\). The relationship between GMRA's and multiwavelets is explored, and as a consequence the authors are able to construct all wavelet sets in \(R^n\), not only the ones arising from a multiresolution analysis. Several new wavelet sets are constructed.