从最优多边形的边缘向量中获取相位恢复的最优框架
Optimal Frames for Phase Retrieval from Edge Vectors of Optimal Polygons
AI总结：
本文研究了相位恢复中从最优多边形边缘向量获取的最优框架，揭示了其与等直径问题的关系，并推翻了关于调和框架最优性的猜想。
摘要：
本文旨在表征相位恢复的最优框架，即条件数达到最小值的框架。在二维实数情况下，我们揭示了相位恢复的最优框架与最大化周长的等直径问题之间的联系，该问题最初由Reinhardt于1922年提出。我们的工作证明了每一个最大化周长的等直径问题的最优解本质上都会导致二维空间中的最优框架。通过将最优多边形问题重新表述为关于单位根的偏差问题，我们表征了所有最优多边形。在此基础上，我们进一步表征了当m≥3且m有奇数因子时，二维空间中具有m个向量的最优框架。作为关键结论，我们证明了当m为偶数且m≥4时，调和框架E_m⊂R²并不最优。这一发现推翻了Xia、Xu和Xu在[Math. Comp.] 94 (2025), pp.~2931--2960中提出的猜想。先前的工作已经证明，当m为奇数时，E_m确实最优。
英文摘要：
This paper aims to characterize the optimal frame for phase retrieval, defined as the frame whose condition number for phase retrieval attains its minimal value. In the context of the two-dimensional real case, we reveal the connection between optimal frames for phase retrieval and the perimeter-maximizing isodiametric problem, originally proposed by Reinhardt in 1922. Our work establishes that every optimal solution to the perimeter-maximizing isodiametric problem inherently leads to an optimal frame in ${\mathbb R}^2$. By recasting the optimal polygons problem as one concerning the discrepancy of roots of unity, we characterize all optimal polygons. Building upon this connection, we then characterize all optimal frames with $m$ vectors in ${\mathbb R}^2$ for phase retrieval when $m \geq 3$ has an odd factor. As a key corollary, we show that the harmonic frame $E_m \subset {\mathbb R}^2$ is {\em not} optimal for any even integer $m \geq 4$. This finding disproves a conjecture proposed by Xia, Xu, and Xu [{\em Math. Comp.}, 94 (2025), pp.~2931--2960]. Previous work has established that $E_m$ is indeed optimal when $m$ is an odd integer.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
