Polynomial Inequalities and Applications

多项式不等式及其应用

• 批准号: 1564541
• 负责人: Vilmos Totik
• 金额: $ 9.75万
• 依托单位: University of South Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1564541&HistoricalAwards=false
• 关键词: Polynomial; Inequalities; Applications

This research project concerns mathematical analysis, in particular in approximation theory, orthogonal polynomials, and potential theory. These are parts of mathematics that originated well over a century ago and have provided the foundations and the tools for widespread applications, from magnetic resonance imaging scanners to airplane designs. One main focus of the work is to understand complicated mathematical objects in terms of relatively transparent and computable quantities. The project is a continuation of this classical area, but with a new and fresh look at some of its questions, aimed at developing new methods to solve some well-known open problems. Though it is basic research, the results of the project are anticipated to be useful in other areas of mathematics, physics, and engineering. It is expected that the project will stimulate interest in undergraduates and enhance the research environment for them. There are four main research areas that are considered. The first is the approximative extensions of conformal maps/Green's functions when conventional conformal extensions do not exist. These can be used in lieu of conformal extensions when less smoothness than analyticity is assumed. The second area concerns new types of polynomial inequalities to settle the best constant problem in Bernstein- and Markov-type inequalities on sets consisting of smooth Jordan curves and arcs, thereby closing a chapter in approximation theory that is more than a century old. The third is the study of Christoffel-Darboux kernels associated with general families of orthogonal polynomials with a possible application to universality results in random matrix theory. The fourth research area concerns the clarification and resolution of Widom's conjecture on the norm of Chebyshev polynomials. Various polynomial inequalities connect these fields and are expected to play a decisive role in the solutions of the problems under study.

这个研究项目涉及数学分析，特别是在近似理论，正交多项式，和势理论。这些是数学的一部分，起源于一个多世纪以前，为从磁共振成像扫描仪到飞机设计等广泛应用提供了基础和工具。这项工作的一个主要焦点是用相对透明和可计算的量来理解复杂的数学对象。该项目是这一经典领域的延续，但对其中的一些问题进行了全新的审视，旨在开发新的方法来解决一些众所周知的开放性问题。虽然这是一项基础研究，但该项目的结果有望在数学、物理和工程等其他领域发挥作用。期望这项计划能激发本科生的兴趣，并改善他们的研究环境。有四个主要研究领域被考虑。首先是在常规保角扩展不存在的情况下，保角映射/格林函数的近似扩展。当光滑性低于分析性时，这些可以用来代替保角扩展。第二个领域涉及新的多项式不等式类型，以解决由光滑约旦曲线和圆弧组成的集上的Bernstein和markov型不等式的最佳常数问题，从而结束了一个多世纪以来的近似理论的一章。第三是研究与一般正交多项式族相关的Christoffel-Darboux核，并可能应用于随机矩阵理论的普适性结果。第四个研究领域是关于切比雪夫多项式范数的威多姆猜想的澄清和解决。各种多项式不等式将这些领域联系起来，并有望在研究问题的解决中发挥决定性作用。