An Operator Based Approach to Irregular Frames of Translates

BALAZS, PETER; HEINEKEN, SIGRID

Mathematics

MDPI

Año: 2019 vol. 7

We consider translates of functions in $L^2(R^d)$ along an irregular set of points, i.e. ${phi( cdot - lambda_k)}_{kin Z}$ - where $phi$ 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 sequence or Riesz sequence.We show the connection of theframe-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.