Completeness Problems in Harmonic Analysis and Spectral Theory

这个项目是在复杂和谐波分析和应用领域的频谱理论和数学物理。它关注的开放问题的完整性复杂的指数加权空间的平方可积函数张贴几十年前等著名的数学家诺曼莱文森，安德烈Kolmogorov，马克克莱因，乔治波利亚和诺伯特维纳。在这个项目中，将重新讨论这一领域的三个关键问题，Beurling-Malliavin问题，差距问题和类型问题。Beruling-Malliavin问题在60年代初得到了解决，尽管它的许多推广和应用仍然是开放的。研究者最近提出了解决差距问题的办法。该解决方案提出了一些将在本项目中考虑的新问题。类型问题，也许是三个问题中研究最多、最重要的一个，仍然是一个悬而未决的问题。最近的部分结果，通过用于差距问题的方法产生，表明一个完整的解决方案可能最终是触手可及的。问题的近似任意波的组合的简单波，或“谐波”，属于非常基础的谐波分析。许多著名的数学家都研究过这个问题。尽管作出了相当大的努力，但其中许多问题仍然悬而未决。这类问题在逼近理论、信号处理、预测理论、微分算子谱理论和数学物理中有许多重要的应用。 在这方面取得进一步进展是该项目的主要目标。这些问题的解决方案可以带来一个令人满意的结束了几十年来的研究，这方面的谐波分析。这些问题中的一些在半个多世纪中仍然是开放的，其中一些甚至被认为是“先验的”，即没有封闭形式的解决方案。然而，提案主要部分所述的最近制定的办法为其中几个问题提供了解决办法，并使该地区出现了新的乐观情绪。最近在得克萨斯农工大学举行了该项目领域的一次研究会议（2011年1月）。该项目的主要课题的调查将包括在研究生复杂的分析类在得克萨斯州A M大学的研究员教。2011年，一个国际研究方案的研究员将在该项目领域为年轻研究人员开设两门微型课程。

This project is in the area of Complex and Harmonic Analysis and applications to Spectral Theory and Mathematical Physics. It concerns open problems on completeness of complex exponentials in weighted spaces of square-integrable functions posted decades ago by such prominent mathematicians as Norman Levinson, Andrei Kolmogorov, Mark Krein, George Polya and Norbert Wiener. Three key problems in this area, the Beurling-Malliavin Problem, the Gap Problem and the Type Problem will be revisited in this project. The Beruling-Malliavin Problem was solved in the early sixties, although its numerous generalizations and applications remain open. A solution to the Gap Problem was recently suggested by the investigator. The solution brings up a number of new questions that will be considered in this project. The Type Problem, perhaps the most studied and important of the three, remains open. Recent partial results, produced via the approach used for the Gap Problem, suggest that a full solution may finally be within reach.The problem of approximating an arbitrary wave by combinations of simple waves, or "harmonics," belongs to the very foundations of Harmonic Analysis. Versions of this problem have been studied by many famous mathematicians. Despite considerable efforts, many of such questions remain open. Such problems have a number of important applications in Approximation Theory, Signal Processing, Prediction Theory, Spectral Theory of differential operators and Mathematical Physics. Further progress in this direction is the main goal of the project. Solutions to these problems could bring a satisfactory closure to the decades-long study of this area of Harmonic Analysis. Some of these problems remained open for more than half a century, and some of them were even considered "transcendental," i.e. not having a closed form solution. However, a recently developed approach, described in the main part of the proposal, has produced solutions to several of such problems and brought new optimism to the area. A research conference in the area of the project was recently held at Texas A&M University (January 2011). A survey of the main topics of the project will be included in the graduate Complex Analysis classes taught by the investigator at Texas A&M University. Two minicourses in the area of the project, directed at young researchers, will be given by the investigator at an international research program in 2011.