Construction of Parseval wavelets from redundant filter systems

We consider wavelets in L 2 ( R d ) which have generalized multiresolutions. This means that the initial resolution subspace V 0 in L 2 ( R d ) is not singly generated. As a result, the representation of the integer lattice Z d restricted to V 0 has a nontrivial multiplicity function. We show how the corresponding analysis and synthesis for these wavelets can be understood in terms of unitary-matrix-valued functions on a torus acting on a certain vector bundle. Specifically, we show how the wavelet functions on R d can be constructed directly from the generalized wavelet filters.