Necessary and sufficient conditions for constructing orthonormal wavelet bases

Abstract

This paper proves a previous conjecture of the author characterizing sequences h∈l²(Z) that yield orthonormal wavelet bases of L²(R) in terms of the multiplicity of the eigenvalue 1 of an operator associated to h. The proof utilizes a result of Cohen characterizing these sequences in terms of the real zeros of their Fourier transforms. The mapping from sequences to wavelets is shown to define a continuous mapping from a subset of l²(Z) into L²(R). Related conjectures are discussed.