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.

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关于一类弱R-对偶和对偶关系

序列的 R 对偶的概念最初是为了在 Gabor 分析中获得对偶原理的一般版本而引入的。从那时起，一些数学家引入并研究了各种 R-对偶（II、III、IV 型）和 R-对偶设置的一些松弛。所有这些“R-对偶”为一般框架理论中对偶关系的分析提供了有力的工具。它对数学具有独立的兴趣，远远超出 Gabor 分析中的对偶原理。观察到 R-对偶的底层序列是一对正交碱基。在本文中，我们介绍了基于一对 Parseval 框架的弱 R-对偶的概念。它是 R-dual 设置的新放松。我们根据弱 R-对偶获得帧的特征，并证明一帧的弱R-对偶（Riesz 基）是一个帧序列（frame）。我们还根据弱 R 对偶表征（统一）等效框架。最后，我们提出了弱 R 对偶的正则对偶的明确表达。