In this note, we study frame-related properties of a sequence of functions multiplied by another function. In particular, frame and Riesz basis properties are obtained which are applied to the sets of irregular translates of a bandlimited function h in  This is achieved by looking at a set of exponentials restricted to a set  with frequencies in a countable set L and multiplying it by the Fourier transform of a fixed function  Using density results due to Beurling, we prove the existence and give ways to construct frames by irregular translates.