Characterizations of weak R-duality and its application to Gabor frames

TL;DR

This paper characterizes when a frame sequence is a weak R-dual of another and applies these results to Gabor frames, showing that weak R-duality of the adjoint Gabor system implies R-duality.

Contribution

It provides new characterization criteria for weak R-duals and connects weak R-duality with R-duality in Gabor frames, advancing understanding of duality relations.

Findings

Characterization of weak R-duals for frame sequences

Weak R-duality of the adjoint Gabor system implies R-duality

Progress towards the problem of the adjoint Gabor system being an R-dual

Abstract

Weak R-duals, a generalization of R-duals, were recently introduced; for which duality relations were established. In this paper, we consider the problem of characterizing a given frame sequence to be a weak R-dual of a given frame. Further, we apply these characterization results to the Gabor frame setting and prove that the weak R-duality of the adjoint Gabor system leads to the R-duality of the same, thereby indicating an approach to answer the famous problem of the adjoint Gabor system being an R-dual of a given Gabor frame.