Potentials, Polynomials and Weighted Polynomial Inequalities

主要研究者探讨了近似理论中的几个问题，对势和多项式的应用特别感兴趣。其中一个主题是变权多项式近似。最近有几个问题导致了这种近似，并且PI在存在外场的情况下借助对数势继续其研究。这个近似问题是证明关于变权的正交多项式和克里斯托费尔函数的不同类型渐近性的基础。这些反过来又与量子物理的一些力学模型有关，特别是与普适性假设有关（即小间隔内能级的分布与粒子的原始（随机）分布无关）。另一个相关的问题是关于复平面上紧支持测度的正交多项式（多项式的界、正则性准则、曲线或不相交区间上的Szego问题）。最后，PI计划研究具有尽可能一般权值的多项式不等式。最近我们发现经典多项式不等式在加权形式下也是成立的，并且对于非常广泛的权重类别，也就是所谓的加倍权重，或者对于满足Muckenhoupt a最弱条件的权重。PI进一步研究了加权多项式不等式，以及它们在其他领域的影响。本课题主要研究数学分析中的一些理论问题，旨在将其应用于数学和理论物理的更多应用领域。用传统上与经典物理（势理论）相关的工具来研究近似理论中的一些基本问题。势理论的这种新应用使得解决经典应用工具无法解决的问题成为可能，例如，它使得发现一些非经典正交多项式（弗洛伊德多项式）的行为成为可能。近年来，人们发现这些正交多项式与理论物理中的一些问题有关，这些问题是用随机力学模型来模拟量子系统的不确定性。另一个主要的研究领域是多项式上的加权不等式。多项式是用可以用数学方法处理的简单结构代替复杂结构的基本工具；在这个近似过程中我们需要不同的不等式将不同数量的多项式彼此联系起来。

The principal investigator explores several problems in approximation theory with special interest in applications of potentials and in polynomials. One of these topics is polynomial approximation with varying weights. Recently several problems have lead to this type of approximation, and the PI continues its study with the help of logarithmic potentials in the presence of an external field. This approximation problem is fundamental in proving different types of asymptotics for orthogonal polynomials and Christoffel functions with respect to varying weights. These in turn are related to some mechanical models of quantum physics, in particular to the universality hypothesis (saying that the distribution of energy levels in small intervals is independent of the original (random) distribution of the particles). Another related problem is on orthogonal polynomials with respect to a measure with compact support on the complex plane (bounds for polynomials, regularity criteria, Szego's problem on curves or disjoint intervals). Finally, the PI plans to investigate polynomial inequalities with as general weights as possible. Recently it has turned out that classical polynomial inequalities are also true in weighted form, and for a surprisingly wide class of weights, namely for so called doubling weights, or for weights satisfying the weakest of the Muckenhoupt A conditions. The PI further investigates weighted polynomial inequalities, as well as their consequences in other areas. This research project deals with some theoretical questions of mathematical analysis with the aim of using in more applied areas of mathematics and theoretical physics. Some basic problems in approximation theory are investigated with tools that have been traditionally associated with classical physics (potential theory). This novel application of potential theory makes it possible to attack problems that resisted classically applied tools, e.g. it makes possible to find the behav ior of some nonclassical orthogonal polynomials (Freud polynomials). A new impulse to the research came in recent years when it has been found that these orthogonal polynomials are related to some questions of theoretical physics that are modelling the uncertain nature of quantum systems by random mechanical models. Another major area of research is on weighted inequalities on polynomials. Polynomials are basic tools in replacing complicated structures by simpler ones that can be mathematically handled; and in this approximation process one needs various inequalities that relate various quantities of polynomials to one another.