近似对偶框架的不存在性：加权复合算子视角
Nonexistence of Approximately Dual Frames via Weighted Composition Operators
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中文总结 AI 辅助
本文研究加权复合算子能否实现框架的对偶或近似对偶，证明在Hardy和Bargmann-Fock空间上，严格对偶迫使算子为恒等映射的正标量倍，而近似对偶在Hardy空间同样刚性，在Bargmann-Fock空间则存在由误差决定的尖锐阈值。
中文摘要 AI 辅助
从框架的系数进行重构通常需要一个对偶框架，而典范对偶是通过对框架算子求逆得到的，该算子很少能以闭式形式获得。因此，自然要问：对偶性是否可以由具有规定形式的算子来实现。我们在Hardy空间和Bargmann-Fock空间上，在加权复合算子的背景下研究此问题。我们首先刻画了那些将一个框架及其一个对偶框架映射到另一对这样的框架的加权复合算子。然后我们证明，如果框架在加权复合算子下的像本身是一个对偶框架，那么该算子必是恒等算子的正标量倍数，并且该框架在Hardy空间和Bargmann-Fock空间上都是紧框架。将对偶性放宽为近似对偶性，我们证明在Hardy空间上，在权函数的温和正则性条件下，复合仍被迫为恒等映射。然而，近似紧性完全失效。在Bargmann-Fock空间上，情况是刚性的：当且仅当最优框架界之比不超过由规定误差确定的显式尖锐阈值时，这样的近似对偶才存在。
英文摘要
Reconstruction from the coefficients of a frame often requires a dual frame, and the canonical dual is obtained by inverting the frame operator, which is rarely available in a closed form. It is therefore natural to ask whether the duality can instead be implemented by an operator of prescribed form. We study this problem in the context of weighted composition operators on Hardy and Bargmann-Fock spaces. We first characterize those weighted composition operators which carry a frame together with a dual of it to another such pair. We then prove that if the image of a frame under a weighted composition operator is itself a dual frame, then such an operator is a positive scalar multiple of the identity and the frame is tight on both Hardy and Bargmann-Fock spaces. Relaxing duality to approximate duality, we show that on the Hardy space, composition is still forced to be the identity under mild regularity for the weight function. Approximate tightness, however, fails completely. On the Bargmann-Fock space, the situation is rigid: such an approximate dual exists precisely when the ratio of the optimal frame bounds does not exceed an explicit sharp threshold determined by the prescribed error.
发表机构
- Dartmouth College（达特茅斯学院）
- University of Idaho（爱达荷大学）
- Georgia Institute of Technology（佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。
