Nonuniform Multiresolution Analyses and Spectral Pairs

AbstractA 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