On R-duals and the duality principle

Abstract

In 2004 Casazza, Kutyniok and Lammers introduced the R-duals of sequences in a general Hilbert space. The purpose was to obtain a general version of the duality principle in Gabor analysis. It was shown that the R-duals cover the duality principle for tight Gabor frames and Gabor Riesz bases. In this paper we discuss the relationship between the R-duals and a variant, called R-duals of type III, introduced in 2014. In contrast to the original R-duals, it is known that the R-duals of type III generalize the duality principle for all Gabor frames, but we believe that a smaller and more convenient class will work as well. The purpose of the paper is to give a focussed presentation of the R-duals of type I and III that can trigger the research on this.