The classical Wiener's Tauberian theorem states that the system of all translations of a function in the Lebesgue space $L^1(R)$ is dense if and only if its Fourier transform is non vanishing on R. A similar characterization is true for the Lebesgue space of square-integrable functions $L^2(R)$. However, this is not true in general Lebesgue space $L^p(R)$. In this talk, we will discuss the completeness property of the system generated by discrete translations of a function. This is joint work with Ms. Bhawna Dharra.