Completeness of Orthonormal Wavelet Systems for Arbitrary Real Dilations

Abstract

It is shown that the discrete Calderón condition characterizes completeness of orthonormal wavelet systems, for arbitrary real dilations. That is, if a>1,b>0, and the system Ψ={aj/2ψ(ajx-bk):j,k∈Z} is orthonormal in L2(R), then Ψ is a basis for L2(R) if and only if ∑j∈Zψ̂(ajξ)2=b for almost every ξ∈R. A new proof of the Second Oversampling Theorem is found, by similar methods. © 2001 Academic Press.