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

通过势论和几何函数理论研究实线子集的构造函数论

• 批准号: 0554344
• 负责人: Volodymyr Andriyevskyy
• 金额: $ 9.86万
• 依托单位: Kent State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-15 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0554344&HistoricalAwards=false
• 关键词: Contructive; Function; Theory; Subsets; Real

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.

摘要：本文的主要目的是研究构成分析数学和应用数学共同基础的构造函数论中的一系列基本问题。我们的方法借鉴了理论数学许多领域的思想和技术，例如真实的和复分析、拓扑学和傅里叶分析。最近，Carleson，Totik和PI在一系列的论文中发现了一种新的方法，将绿色函数的连续性和定义了绿色函数的区域的边界的度量性联系起来。我们相信，这种方法可以给一个决定性的推动调查的一些长期存在的开放性问题，在constructivefunctio理论。这个建议的一个主要组成部分是研究一个新的表示的基本概念的潜在的理论（对数容量，绿色函数，和平衡测度）用单位区间的外部到具有有限个或无限个径向狭缝的单位圆盘的外部的保角映射表示，在PI的最近的工作中提出。我们分析康托的几何形状-型集，并对Totik和Carleson关于具有Hoelder连续绿色函数的起始集的结果给出了新的证明和重要的推广.第二部分我们的建议涉及马尔可夫和Remez型不等式多项式子集的真实的线。我们提出了一个广义的2维Remez型不等式理论，并通过一些应用说明了它们的威力.最后一部分研究了复平面上多项式逼近中的一些著名问题，这些问题在理论数学和应用数学中都有着广泛的应用。我们希望在多项式的倒数在正真实的轴上几何收敛于函数的倒数的情况下，找到关于整函数结构的Meinardus-Varga问题的完全解。我们打算采用一个新的概念的Faber型多项式。我们以前的研究表明，Nikolskiii-Timan-Dzjadyk逼近定理与Beardon和Pommerenke引入的一致完全集概念之间存在着联系。我们建议调查这种联系的细节。我们的建议的更广泛的影响的主要组成部分是创建新的潜在的理论，几何函数理论和建设性的函数理论之间的联系。另一个组成部分涉及研究生和本科生的培训。事实上，这个建议中所涉及的问题是以这样一种方式陈述的，不仅研究生清楚，而且本科生也可以理解。另一方面，这些问题的答案是相当反直觉的。这激发了学生对这门学科和数学的兴趣.