Contructive Function Theory on Subsets of the Real Line Through Potential Theory and Geometric Function Theory

摘要：本文的主要目的是研究构造函数理论中的一系列基本问题，这些问题构成了分析数学和应用数学的共同基础。我们的方法借鉴了许多理论数学领域的思想和技术，如实和复分析、拓扑和傅立叶分析。最近，Carleson，Totik和PI在一系列论文中发现了一种新的方法来连接Green函数的连续性质和定义Green函数的定域边界的度量性质。我们相信，这种方法可以对构造函数理论中一些长期存在的开放性问题的研究起到决定性的推动作用。这个建议的一个主要组成部分是研究在PI最近的工作中提出的单位区间的外部到具有有限或无限数量径向狭缝的单位圆盘的外部的保角映射方面的势理论基本概念（对数容量，格林函数和平衡测度）的新表示。我们分析了cantor型集合的几何性质，提出了Totik和Carleson集合具有Hoelder连续Green函数的新证明和对结果的重要推广。我们的建议的第二部分是关于实线的多项式子集的马尔可夫和雷米兹型不等式。我们建议构造一个一般性的$2$维remez型不等式理论，并通过给出一些应用来说明它们的力量。最后一部分研究了复平面上多项式近似的一些著名的开放问题，这些问题在纯数学和应用数学中都有大量的应用。我们希望找到一个完整函数的Meinardus-Varga问题结构的完全解，该结构在多项式的倒数正实轴上收敛于函数的倒数。我们打算采用费伯多项式的新概念。我们先前的研究表明nikolskii - timan - dzjadyk近似定理与Beardon和Pommerenke提出的一致完美集概念之间存在联系。我们建议调查这一联系的细节。我们建议的更广泛影响的主要组成部分是在势理论，几何函数理论和构造函数理论之间建立新的联系。另一个组成部分涉及研究生和本科生的培养。事实上，本提案中所讨论的问题不仅对研究生来说很清楚，而且对本科生来说也很容易理解。另一方面，许多问题的答案都是与直觉相反的。这激发了学生对这门学科和数学的兴趣。

ABSTRACT:The main goal of this proposal is to investigate a series of fundamentalproblems in constructive function theory which constitute common groundof the analysis and applied mathematics. Our approach borrows ideas andtechniques from many fields of theoretical mathematics, such as real andcomplex analysis, topology, and Fourier analysis. Recently, Carleson,Totik, and the PI in a series of papers have found a new approach toconnect the continuous properties of the Green function and the metricproperties of the boundary of a domain where the Green function isdefined. We believe that this approach can give a decisive impulse toinvestigation of a number of long-standing open problems in constructivefunctio theory. A major component of this proposal is to study a newrepresentation of basic notions of potential theory (logarithmic capacity, the Green function, andequilibrium measure) in terms of a conformal mapping of the exterior ofthe unit interval onto the exterior of the unit disk with finite orinfinite number of radial slits, presented in the recent work of the PI.We analyze the geometry of Cantor-type sets and propose to find a newproof to and significant extension of the results by Totik and Carleson onsets possessing the Hoelder continuous Green function. The second part ofour proposal concerns Markov- and Remez-type inequalities for polynomialson subsets of the real line. We propose to construct a general$2$-dimensional theory of Remez-type inequalities and illustrate theirpower by giving a number of applications. The last part is devoted to study of well-knownopen problems in polynomial approximation in the complex plane which havea large number of applications in both pure and applied mathematics. Wehope to find a complete solution of the Meinardus-Varga problem onstructure of an entire function with the geometric convergence on thepositive real axis of reciprocals of polynomials to the reciprocal of thefunction. We intend to employ a new concept of Faber-type polynomials. Ourprior reseach indicates that there exists a connection between thNikolskii-Timan-Dzjadyk approximation theorem and the concept of uniformlyperfect sets introduced by Beardon and Pommerenke. We propose toinvestigate the details of this connection. The major component of the broader impact of our proposal is to create anew link between potential theory, geometric function theory andconstructive function theory. Another component concerns the trainingof graduate and undergraduate students. Indeed, the problems addressed inthis proposal are stated in such a way that not only are they clear to thegraduate students but they are accessible to undergraduates as well. Onthe other hand, the answers to many of those problems are quitecounterintuitive. This stimulates the interest of students to the subjectand mathematics in general.