Approximation by translates and completeness of weighted systems of exponential functions in $L^2(\mathbf R)$

Both necessary conditions and sufficient conditions on a sequence $\lambda_n\in\mathbf R$ are found for the family of translates $f(t-\lambda_n)$ of an $L^2$-function whose Fourier transform is almost everywhere nonzero and rapidly decreasing to be dense in $L^2(\mathbf R)$.
Bibliography: 9 titles.