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

研究将在无限连通域上的调和测度的度量性质，与多项式不等式和正交多项式。调和测度/绿色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给出。 这样的结果直接导致估计多项式在复平面上和一些经典的多项式不等式一般紧集的扩展。特别是，这些结果将允许找到康托型集的马尔可夫因子的阶，这反过来将产生零测度的紧集，其马尔可夫因子具有最小增长阶O（n^2）。关于一般（比如加倍）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。另一部分研究是多项式逼近和变权正交多项式（权随次数而变），这将在外场存在下借助对数势进行研究。近似问题是证明变权Christoffel函数和正交多项式的渐近性的基础，而这些渐近性又可以用来检验量子物理力学模型中的普适性假设。本研究所使用的工具与经典的分析方法不同，但主要是势理论的方法，并将以数学分析的方法进行研究。主要的努力将是寻找新的方法来估计一些经典的数量和推广一些众所周知的结果更一般的情况。这些将允许更广泛的applicabilityof几个经典的工具，并将提供一些新的见解的理论方面的一些基本问题的逼近theoryand正交多项式。

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.