  In this note we study frame-related properties of a sequence of functions
multiplied by another function. In particular we study frame and Riesz basis
properties. We apply these results to sets of irregular translates of a
bandlimited function $h$ in $L^2(\R^d)$. This is achieved by looking at a set
of exponentials restricted to a set $E \subset \R^d$ with frequencies in a
countable set $\Lambda$ and multiplying it by the Fourier transform of a fixed
function $h \in L^2(E)$. Using density results due to Beurling, we prove the
existence and give ways to construct frames by irregular translates.
