Polynomial Inequalities and Applications

这个研究项目涉及数学分析，特别是近似理论，正交多项式和潜在的理论。这些都是数学的一部分，起源于世纪以前，并为广泛的应用提供了基础和工具，从磁共振成像扫描仪到飞机设计。这项工作的一个主要重点是理解复杂的数学对象相对透明和可计算的数量。该项目是这一经典领域的延续，但对其中的一些问题有了新的看法，旨在开发新的方法来解决一些众所周知的开放问题。虽然这是基础研究，但该项目的结果预计将在数学，物理和工程的其他领域有用。预计该项目将激发本科生的兴趣，并改善他们的研究环境。有四个主要的研究领域被认为是。第一个是当传统的共形扩张不存在时，共形映射/绿色函数的近似扩张。这些可以用来代替保形扩张时，少光滑比解析性是假定。第二个领域涉及新类型的多项式不等式，以解决最好的常数问题，在伯恩斯坦和马尔可夫型不等式的集合组成的光滑约旦曲线和弧，从而关闭了一章近似理论是超过一个世纪的老。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中的普遍性结果。 第四个研究领域涉及Widom关于Chebyshev多项式范数的猜想的澄清和解决。各种多项式不等式连接这些领域，并预计将发挥决定性的作用，在研究中的问题的解决方案。

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.