消失矩膨胀及其在平衡框架与图框架中的应用

中文摘要 AI 辅助

给定矩阵 $A\in M_{k\times N}(\Bbb{F})$，本文研究具有消失 $A$-矩的有限框架 $\{f_j\}_{j=1}^N$，并探讨其与框架膨胀性质的联系。我们建立了消失 $A$-矩膨胀存在性的若干等价刻画，包括涉及对偶框架、算子值域及与矩阵 $A$ 相关的线性独立性性质的条件。特别地，我们证明每个冗余度为 $k$ 的框架，对某个完全非奇异矩阵 $A$，都存在具有消失 $A$-矩的对偶框架。由此得到的一个关键结论是：通过将此类对偶框架用作编码框架，任何冗余度为 $k$ 的框架（无需具备删除鲁棒性），均可对任意数量的 $\ell$ 次删除（$\ell \leq k$）实现完美信号重构，且仅需对大小为 $\ell$ 的小矩阵求逆，而非对大小为 $N-\ell$ 的子框架算子（大矩阵）求逆。这类框架的一个特例是 $k$ 阶消失矩框架，其与图框架存在自然联系。我们通过多项式消去恒等式和算子理论条件，刻画了具有 $k$ 阶消失矩的对偶框架的存在性。我们进一步讨论了其在删除恢复和图框架问题中的若干应用。这些结果为连接消失矩条件、对偶框架理论、膨胀理论及框架的图论构造提供了统一框架。

英文摘要

Given a matrix $A\in M_{k\times N}(\Bbb{F}),$ we examine finite frames $\{f_j\}_{j=1}^N$ with vanishing $A$-moments and investigate their connections with the frame dilation property. We establish several equivalent characterizations for the existence of vanishing $A-$moment dilations, including conditions involving dual frames, operator ranges, and linear independence properties associated with the matrix $A$. In particular, we show that every frame with redundancy $k$ admits a dual frame with vanishing $A$-moments for some totally nonsingular matrix $A$. A key takeaway of this is that by invoking such a dual frame as the encoding frame, any frame with redundancy $k$ (erasure robustness is not needed) allows perfect signal reconstructions for any number of $\ell$-erasures ($\ell \leq k$) which only involves inverting a matrix of small size ($\ell$) instead of inverting the subframe operator which is a matrix of large size ($N-\ell$). A special case of this type of frames are $k$-th vanishing moment frames which have natural connections with graph frames. We characterize the existence of dual frames with $k-th$ vanishing moments through polynomial cancellation identities and operator theoretical conditions. We further discuss some applications to problems in erasure recovery and graph frames. These results provide a unified framework linking vanishing moment conditions, dual frame theory, dilation theory, and graph theoretic constructions of frames.

发表机构

• University of Central Florida（中佛罗里达大学）

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