# Criteria for generalized translation-invariant frames

## TL;DR

This paper establishes new Fourier domain criteria for generalized translation-invariant frames, incorporating phase considerations to improve the characterization of frame properties, especially for tight frames, in various settings.

## Contribution

It introduces phase-aware Fourier conditions for translation-invariant systems, extending known results to locally compact abelian groups and wavelet systems.

## Key findings

- Phase considerations improve frame characterization.
- Conditions are optimal for tight frames.
- Results apply to Euclidean space and wavelet systems.

## Abstract

This paper provides new sufficient and necessary conditions for the frame property of generalized translation-invariant systems. 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 results on generalized translation-invariant systems will be proved in the setting of locally compact abelian groups, but even for euclidean space and the special case of (composite) wavelet systems the results are new.