Two Conjectures on Finite Gabor Systems

数字信号处理的进步往往依赖于纯谐波分析和计算谐波分析的成功，谐波分析是数学的一个分支，可以追溯到约瑟夫·傅立叶。这些成功的例子包括引入用于压缩摄影图像的JPEG标准，在无相位重建方面取得进展，以及压缩感知，例如，单像素相机的构造。这一进步的核心是更好地理解我们日常生活中产生的许多数据中固有的冗余。框架理论可以看作是研究和建模冗余的一种合适的范式。研究者利用框架理论研究了两类问题，这两类问题的解在量子信息理论中可能具有重要的适用性。由于本项目中涉及的一些数学问题与工程问题有关，它们的解决方案可能会导致信号处理和技术基础设施的进步，并拓宽对框架在应用中的理解和作用。研究了量子信息论中的Zauner猜想和时频分析中的HRT猜想。他们观察到Zauner猜想是HRT猜想在秩一有限维时频矩阵设置中的一个特殊情况，并且最初将这种关系用于每个猜想的当前技术转移。框架理论在形成和理解研究者在这里考察的问题方面起着根本性的作用。有限向量集的相干性的概念是在解决这些问题中取得技术进步所必需的一个重要的定量度量，特别是在理解这些集合可能具有的最大不相干性的作用方面。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Advances in digital signal processing often rely on successes in pure and computational harmonic analysis, a branch of mathematics dating back to Joseph Fourier. Examples of these successes include the introduction of the JPEG standard for compression of photographic images, advances in phaseless reconstruction, and compressed sensing, e.g., the construction of single pixel cameras. At the core of this progress is a better understanding of the redundancy inherent in many data generated in our daily lives. Frame theory can be viewed as one of the appropriate paradigms to investigate and model redundancy. The investigators use frame theory to study two classes of problems whose solutions could have significant applicability in quantum information theory. Because some of the mathematical problems taken up in this project are related to engineering problems, their solutions could lead to advances in signal processing and technological infrastructure, as well as broaden the understanding and role of frames in applications.The investigators study the Zauner conjecture in quantum information theory and the HRT (Heil-Ramanathan-Topiwala) conjecture in time-frequency analysis. They observe that the Zauner conjecture is a special case of the HRT conjecture in the setting of rank-one finite-dimensional time-frequency matrices, and use that relationship initially for the transference of current technology for each conjecture. The theory of frames plays a fundamental role in formulating and understanding the problems the investigators examine here. The notion of the coherence of finite sets of vectors is an important quantitative measure necessary to make technical progress in solving these problems, especially as regards understanding the role of maximal incoherence that such sets may have.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.