Construction of Parseval wavelets from redundant filter systems

Abstract

We consider wavelets in L2(Rd) which have generalized multiresolutions. This means that the initial resolution subspace V0 in L2(Rd) is not singly generated. As a result, the representation of the integer lattice Zd restricted to V0 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 Rd can be constructed directly from the generalized wavelet filters.