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

本文将研究无穷连通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。谐波测度/格林函数的估计将通过涉及与域边界相关的密度函数的局部积分给出。这些结果直接导致了复数平面上多项式的估计和一些经典多项式不等式在一般紧集上的推广。特别是，结果将允许找到康托尔型集的马尔可夫因子的阶数，这反过来将产生零测度的紧集，其中马尔可夫因子的最小增长阶为O（n^2）。关于一般（如加倍）权的加权多项式不等式也将与近似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近似和变权正交多项式（权随程度变化），这将在外场存在的对数电位的帮助下进行研究。逼近问题是证明Christoffel函数和变权正交多项式的适当渐近性的基础，可用于检验量子物理统计力学模型中的普适性假设。本研究使用了不同于经典分析的工具，但主要方法是潜在理论。研究将在数学分析方面进行。主要的努力将是寻找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。这将允许一些经典工具的更广泛的适用性，并将为逼近理论和正交多项式中一些基本问题的理论方面提供一些新的见解。

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.