Nonuniform Multiresolution Analyses and Spectral Pairs

A generalization of the notion of multiresolution analysis, based on the theory of spectral pairs, is considered. In contrast to the standard setting, the associated subspaceV0ofL2(R) has, as an orthonormal basis, a collection of translates of the scaling functionφof the form {φ(x−λ)}λ∈ΛwhereΛ={0, r/N}+2Z,N⩾1 is an integer, andris an odd integer with 1⩽r⩽2N−1 such thatrandNare relatively prime andZis the set of all integers. Furthermore, the corresponding dilation factor is 2N, the case whereN=1 corresponding to the usual definition of a multiresolution analysis with dilation factor 2. A necessary and sufficient condition for the existence of associated wavelets, which is always satisfied whenN=1 or 2, is obtained and is shown to always hold if the Fourier transform ofφis a constant multiple of the characteristic function of a set.