"Criteria and the local integrability condition for generalised translation-invariant frames."
Van Velthoven, JordyNew sufficient and necessary conditions for the frame property of generalised translation-invariant systems are presented. The conditions are formulated in the Fourier domain and consists of estimates involving the upper and lower frame bound. Contrary to known conditions of a similar nature, the estimates take the phase of the generating functions in consideration and not only their modulus. The possibility of phase cancellations makes these estimates optimal for tight frames. The frame criteria are valid under a technical condition known as the local integrability condition (LIC). Simplified conditions on the lattice systems and the generating functions for satisfying the LIC are given.