On various R-duals and the duality principle

TL;DR

This paper investigates the relationship between R-duals and the duality principle in frame theory, proposing a modified R-dual concept that generalizes the duality principle while preserving its key properties.

Contribution

It introduces a modified R-dual framework that extends the duality principle to more general settings, maintaining desirable properties of R-duals.

Findings

A modified R-dual version generalizes the duality principle.

Characterization of R-duals for Riesz bases.

Insight into the relations between sequences and their R-duals.

Abstract

The duality principle states that a Gabor system is a frame if and only if the corresponding adjoint Gabor system is a Riesz sequence. In general Hilbert spaces and without the assumption of any particular structure, Casazza, Kutyniok and Lammers have introduced the so-called R-duals that also lead to a characterization of frames in terms of associated Riesz sequences; however, it is still an open question whether this abstract theory is a generalization of the duality principle. In this paper we prove that a modified version of the R-duals leads to a generalization of the duality principle that keeps all the attractive properties of the R-duals. In order to provide extra insight into the relations between a given sequence and its R-duals, we characterize all the types of R-duals that are available in the literature for the special case where the underlying sequence is a Riesz basis.