Democratic systems of translates

Democratic systems arise in the context of greedy approximations in Banach spaces. Systems of translates of a single function are the basis of shift invariant subspaces and are used in the construction of wavelets and Gabor systems. In this article, we study the democracy in L2(R) of the system of integer translates of a single function ψ∈L2(R). Necessary and sufficient criteria are given in terms of properties of ψ. The problem of finding an (operative) necessary and sufficient condition is still unsolved.