An operator based approach to irregular frames of translates

Abstract

We consider translates of functions in L 2 (Rd ) along an irregular set of points, that is, {φ(· − λk )}k∈Z—where φ is a bandlimited function. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. This is used, in particular, to find for the first time in the irregular case a representation of the canonical dual as well as of the equivalent Parseval frame—in terms of its Fourier transform.