Almost Integer Translates. Do Nice Generators Exist?

Abstract

It has been shown earlier by the first author that for any nonzero perturbation of the integers $\lambda_n=n+o(1), \lambda_n\ne n$, there is a \textit{generator,} that is a function $\varphi\in L^2(\mathbf{R})$ such that the system of translates $\{\varphi(x-\lambda_n)\}$ is complete in $L^2(\mathbf{R})$. We ask if $\varphi$ can be chosen with fast decay. We prove that in general it cannot. On the other hand, if the perturbations are ‘quasianalytically small,’ than it can, and this decay restriction is sharp. A certain class of complex measures which we call ‘shrinkable’ is introduced, and it is shown that the zeros sets of such measures do dot admit generators with fast decay.