Sampling theory and bases of exponentials: novel techniques in the foundations of communications

采样理论是数学研究的一个充满活力的领域，它继续在应用科学和电气工程中发挥核心作用。 近年来，经典抽样理论和指数基框架理论都取得了重要进展。为此，开发了新的建设性方法，并应用了诸如Kadison-Singer问题的突破性解决所产生的功能分析见解。该项目建立在这些新结果的基础上，其中包括新近获得的关于Paley-Wiener空间上的有限区间和/或有限测度集上的Riesz基和指数框架的性质，在非平方可积傅立叶变换函数的Paley-Wiener空间中，Nyquist重构公式缺乏一致收敛性。我们寻求Riesz基地和框架的指数与其他属性和完善的结果是基于积极的解决方案的Kadison-Singer问题。我们还寻求获得今天可用的存在结果的建设性版本。 此外，我们扩展结果缺乏一致收敛的Nyquist重建公式不规则采样公式，例如，开发的Papoulis。我们期望本提案中的多个相关研究焦点将导致显著的交叉施肥。本文考虑的问题涉及通信工程的基础，并与当前数据传输任务直接相关。 例如，子空间的巴利安-洛定理表明，避免临界密度并不能完全缓解巴利安-洛现象。这可能对第五代移动的通信具有影响，在第五代移动通信中，采用诸如正交频分复用的时间-频率传输方法。我们以这些考虑为指导，并希望从我们与从事采样理论和通信的电气工程师的合作中受益。

Sampling theory is a vibrant area of mathematical investigation and it continues to play a central role in applied sciences and electrical engineering. In recent years, fundamental progress has been achieved in the classical sampling theory and the theory of exponential bases and frames. To this end, new constructive methods were developed and functional analytic insights such as those stemming from the groundbreaking resolution of the Kadison-Singer problem were applied.This project builds on these new results, which include newly obtained properties of Riesz bases and frames of exponentials on Paley-Wiener spaces on finite unions of intervals and/or on finite measure sets, and the lack of uniform convergence of the Nyquist reconstruction formula in Paley-Wiener spaces of functions with non-square-integrable Fourier transforms. We seek Riesz bases and frames of exponentials with additional properties and refine results that are based on the positive solution of the Kadison-Singer problem. We also seek to obtain constructive versions of the existence results that are available today. Furthermore, we extend results on the lack of uniform convergence of Nyquist reconstruction formulas to irregular sampling formulas, for example, those developed by Papoulis. We expect that multiple related research foci in this proposal will lead to significant cross fertilization.The questions considered herein address the foundations of communications engineering and are directly relevant for current tasks in data transmission. For example, the Balian-Low theorem for subspaces shows that avoiding critical density does not fully mitigate the Balian-Low phenomenon. This may have impact on the fifth generation mobile communications where time-frequency transmission methods such as orthogonal frequency division multiplexing are employed. We are guided by these considerations and hope to benefit from our collaboration with electrical engineers working in sampling theory and communications.