Discrete Problems in Harmonic Analysis and One Bit Sensing

一方面，该项目将促进人们对信号处理中当前重要主题的理解，另一方面，它将刺激数学，分析和数论两个中心领域之间的相互作用。在信号处理中，所谓的一位感测通过结合测量的符号来取代信号的通常线性测量。这样做有两个动机。 首先，符号函数是非线性的一种基本类型，因此关于它的结果可以代表在许多重要设置中放松线性的第一步。第二，在重要的技术发展中，一位测量可以非常频繁地进行。令人惊讶的实验正在使用需要理论证明的新方法，在这种情况下，一位测量已被证明与线性测量一样有效。 压缩感知中的主题与差异理论中的问题有着密切的关系，一位感知是揭示这种关系的最简单的地方。 这在差异理论中提出了一系列新的问题，并为压缩感知带来了新的技术。 特别是，有新的工具来研究该主题的下界，这可以揭示当前随机技术的有效性。同时，它还可以提供新的采样方法，如半随机抖动采样。 在分析和数论中，该项目将研究最大振荡奇异积分的离散变量。后者，在连续的情况下，已经是迷人的对象。 传递到离散变量需要许多额外的复杂性，这些复杂性来自于微妙的算术结构。 在该项目中建立的结果将是第一个这种类型的，因此将需要新的技术。 这种新方法综合了布尔甘方法和元素方法在相平面分析中的突破性成果。

On the one hand, this project will advance one's understanding of currently important topics in signal processing, and on the other, it will stimulate interactions between two central areas of mathematics, analysis and number theory. In signal processing, so-called one-bit sensing replaces the usual linear measurements of a signal by incorporating the sign of the measurement. There are two motivations for this. First, the sign function is a basic type of nonlinearity, so results about it can represent a first step towards relaxing linearity in a number of significant settings. Second, in an important technological development, one-bit measurements can be done very frequently. Surprising experiments are using new methods that need theoretical justification, and in this context one-bit measurements have proven to be just as effective as linear ones. Topics in compressive sensing have an intimate relationship to questions in discrepancy theory, with one-bit sensing being the easiest place to uncover this relationship. This raises a new set of questions in discrepancy theory, and brings to compressive sensing new techniques. In particular, there are new tools to study lower bounds in the subject, which can reveal how effective the current random techniques are. Also, it can provide new approaches to sampling, such as the semi-random jittered sampling. In analysis and number theory, the project will study discrete variants of maximal oscillatory singular integrals. The latter, in the continuous case, are already fascinating objects. Passing to the discrete variants entails many additional complications that arise from subtle arithmetic structures. The results established in the project will be the first of this type, hence will require novel techniques. The new approach is a synthesis of breakthrough results of Bourgain and elemental methods in phase-plane analysis.