Harmonic measures, potentials, approximation and polynomials inequalities on general sets

一般集上的调和测度、势、近似和多项式不等式

• 批准号: 0097484
• 负责人: Vilmos Totik
• 金额: $ 6.75万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-06-01 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0097484&HistoricalAwards=false
• 关键词: Harmonic; measures; potentials; approximation; polynomials

Research will be done in connection with metric properties of harmonic measures on infinitely connected domains, in connection with polynomial inequalities and orthogonal polynomials. The estimates for harmonic measures/Green functionswill be given via a local integral involving a density function associated withthe boundary of the domain. Such results directly lead to estimates for polynomials in the complex plane and to extensions of some classical polynomial inequalities for general compact sets. The results, in particular, will allow to find the order of the Markoff factors for Cantor-type sets, which in turnwill produce compact sets of zero measure for which the Markoff factors are of the minimal growth order O(n^2). Weighted polynomial inequalitieswith respect to general (say doubling) weights will also be studied together with applications in connection with approximation theory and orthogonal polynomials.Another part of the research is polynomial approximation and orthogonalpolynomials with varying weights (the weight varies with the degree), whichwill be studied with the help of logarithmic potentials in thepresence of an external field. The approximation problem is fundamentalin proving appropriate asymptotics for Christoffel functionsand orthogonal polynomials with varying weights, that inturn can be used to test the universality hypothesis in statistical-mechanical models of quantum physics. The proposed research uses different tools from classical analysis, but the main method is potential thoretic.Research will be done in mathematical analysis. The main effort will be to find new ways to estimate some classical quantities and to extend some well known results to more general situations. These will allow wider applicabilityof several classical tools and will provide some new insights into the theoretical aspects of some fundamental questions in approximation theoryand orthogonal polynomials.

我们将研究无穷连通区域上调和测度的度量性质，以及多项式不等式和正交多项式。调和度量/格林函数的估计将通过涉及与区域边界相关的密度函数的局部积分来给出。这些结果直接给出了复平面上多项式的估计，并推广了一般紧集上的一些经典多项式不等式。特别是，这些结果将允许找到Cantor型集的Markoff因子的阶，这反过来将产生零测度的紧致集，其Markoff因子是O(n^2)的最小增长阶。关于一般(例如倍增)权的加权多项式不等式也将与逼近理论和正交多项式的应用一起研究。研究的另一部分是多项式逼近和具有变权(权随次数变化)的正交多项式，它们将在外场存在的情况下借助于对数势来研究。近似问题是证明Christoffel函数和具有变权的正交多项式的适当渐近性的基础，而这反过来又可以用来检验量子物理统计力学模型中的普适性假设。这项研究使用了与经典分析不同的工具，但主要方法是势能理论。研究将在数学分析中进行。主要的努力将是找到新的方法来估计一些经典的量，并将一些众所周知的结果推广到更一般的情况。这将使几个经典工具具有更广泛的适用性，并将为逼近理论和正交多项式中的一些基本问题的理论方面提供一些新的见解。