An operator based approach to irregular frames of translates

Abstract: We consider translates of functions in $L^2(\RRd)$ along an irregular set of points. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel sequence or Riesz sequence. We find a representation of the canonical dual of a frame sequence ${\phi( \cdot - \lambda_k)}_{k\in \ZZ}$ - where $\phi$ is a bandlimited function - in terms of its Fourier transform.