Abstract
Since the introduction of R-duals by Casazza, Kutyniok and Lammers with the motivation to obtain a general version of duality principle in Gabor analysis, various R-duals and some relaxations of the R-dual setup have been introduced and studied by some mathematicians. They provide a powerful tool in the analysis of duality relations in general frame theory, and are far beyond the duality principle in Gabor analysis. In this paper, we introduce the concept of generalized weak R-dual based on a pair of frames which is a relaxation of the R-dual setup. Using generalized weak R-duals, we characterize the frame properties of a sequence and the equivalence between two frames, prove that the generalized weak R-duals of frames (Riesz bases) are frame sequences (frames), and present a coefficient expression corresponding to the canonical duals of generalized weak R-duals. Some examples are provided to illustrate the generality of the theory.
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