关于各种R-R和对偶原理
On various R-duals and the duality principle
摘要：
对偶原理指出一个Gabor系统是一个框架当且仅当对应的伴随Gabor系统是一个Riesz序列。在一般的希尔伯特空间中，没有任何特殊结构的假设，Casazza，Kutyniok和Lammers引入了所谓的R-矩阵，这也导致了框架在相关Riesz序列方面的特征;然而，这个抽象理论是否是对偶原理的推广仍然是一个悬而未决的问题。在本文中，我们证明了一个修改版本的R-矩阵导致推广的对偶原理，保持所有的吸引力的性质的R-矩阵。为了提供额外的洞察一个给定的序列和它的R-矩阵之间的关系，我们描述了所有类型的R-矩阵，可在文献中的特殊情况下，潜在的序列是一个Riesz基础。
英文摘要：
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.
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