Frames by Multiplication

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.