On a class of weak R-duals and the duality relations

Abstract

The concept of R-duals of a sequence was first introduced with the motivation to obtain a general version of duality principle in Gabor analysis. Since then, various R-duals (types II, III, IV) and some relaxations of the R-dual setup have been introduced and studied by some mathematicians. All these “R-duals” provide a powerful tool in the analysis of duality relations in general frame theory. It is of independent interest in mathematics and far beyond the duality principle in Gabor analysis. Observe that the underlying sequences of a R-dual are a pair of orthonormal bases. In this paper we introduce the concept of weak R-duals based on a pair of Parseval frames. It is a new relaxation of the R-dual setup. We obtain a characterization of frames based on their weak R-duals, and prove that the weak R-dual of a frame (Riesz basis) is a frame sequence (frame). We also characterize (unitarily) equivalent frames in terms of weak R-duals. Finally, we present an explicit expression of the canonical duals of weak R-duals.