Potentials, Polynomials and Weighted Polynomial Inequalities

势、多项式和加权多项式不等式

基本信息

  • 批准号:
    9801435
  • 负责人:
  • 金额:
    $ 6.35万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    1998
  • 资助国家:
    美国
  • 起止时间:
    1998-06-01 至 2002-05-31
  • 项目状态:
    已结题

项目摘要

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.
主要研究人员探讨了几个问题的逼近理论与特别感兴趣的应用潜力和多项式。这些话题之一 是变权的多项式近似。最近 几个问题导致了这种类型的近似, PI在外场存在下借助对数势继续其研究。这个近似问题是基本的证明不同类型的渐近正交多项式和克里斯托弗函数相对于不同的权重。 这些又与量子物理学的某些力学模型有关,特别是与普适性假设有关(即能级在小区间内的分布与粒子的原始(随机)分布无关)。另一个相关的问题是正交多项式相对于一个措施与紧凑的支持在复杂的平面(界限多项式,正则性标准,Szego的问题曲线或不相交的间隔)。最后,PI计划研究具有尽可能一般权重的多项式不等式。最近它已经证明,经典的多项式不等式也是真实的加权形式,并为一个令人惊讶的广泛类的重量,即所谓的加倍重量,或重量满足最弱的Muckenhoupt条件。PI进一步研究加权多项式不等式,以及它们在其他领域的后果。 本课题研究数学分析中的一些理论问题,目的是将其应用于数学和理论物理的更多应用领域。近似理论中的一些基本问题与传统上与经典物理学(势理论)相关的工具进行了研究。势理论的这种新颖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,它使得找到一些非经典正交多项式(弗洛伊德多项式)的近似值成为可能。近年来,当人们发现这些正交多项式与理论物理的一些问题有关时,研究出现了新的推动力,这些问题正在通过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的加权不等式。多项式是替换 复杂的结构,可以用数学处理的简单结构;在这个近似过程中,需要各种不等式,这些不等式将各种数量的多项式相互联系起来。

项目成果

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Vilmos Totik其他文献

Bernstein-type inequalities
  • DOI:
    10.1016/j.jat.2012.03.002
  • 发表时间:
    2012-10-01
  • 期刊:
  • 影响因子:
  • 作者:
    Vilmos Totik
  • 通讯作者:
    Vilmos Totik
The size of irregular points for a measure
  • DOI:
    10.1007/s10474-011-0177-0
  • 发表时间:
    2011-11-29
  • 期刊:
  • 影响因子:
    0.600
  • 作者:
    Vilmos Totik
  • 通讯作者:
    Vilmos Totik
Weighted polynomial inequalities
  • DOI:
    10.1007/bf01893420
  • 发表时间:
    1986-12-01
  • 期刊:
  • 影响因子:
    1.200
  • 作者:
    Paul Nevai;Vilmos Totik
  • 通讯作者:
    Vilmos Totik
Remarks on a functional equation
关于一个函数方程的注记
  • DOI:
    10.14232/actasm-015-805-1
  • 发表时间:
    2015-12-01
  • 期刊:
  • 影响因子:
    0.600
  • 作者:
    Zoltán Daróczy;Vilmos Totik
  • 通讯作者:
    Vilmos Totik
Smooth equilibrium measures and approximation
  • DOI:
    10.1016/j.aim.2006.11.001
  • 发表时间:
    2007-07-10
  • 期刊:
  • 影响因子:
  • 作者:
    Vilmos Totik;Péter P. Varjú
  • 通讯作者:
    Péter P. Varjú

Vilmos Totik的其他文献

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{{ truncateString('Vilmos Totik', 18)}}的其他基金

Polynomial Inequalities and Applications
多项式不等式及其应用
  • 批准号:
    1564541
  • 财政年份:
    2016
  • 资助金额:
    $ 6.35万
  • 项目类别:
    Standard Grant
Harmonic measures in approximation and orthogonal polynomials
近似和正交多项式中的谐波测量
  • 批准号:
    1265375
  • 财政年份:
    2013
  • 资助金额:
    $ 6.35万
  • 项目类别:
    Standard Grant
Christoffel functions and applications
Christoffel 的功能和应用
  • 批准号:
    0968530
  • 财政年份:
    2010
  • 资助金额:
    $ 6.35万
  • 项目类别:
    Standard Grant
Harmonic measures, polynomial inequalities, orthogonal polynomials and approximation
调和测量、多项式不等式、正交多项式和近似
  • 批准号:
    0700471
  • 财政年份:
    2007
  • 资助金额:
    $ 6.35万
  • 项目类别:
    Standard Grant
Smoothness Properties of Harmonic Measures and the Polynomial Inverse Image Method
谐波测度的平滑特性和多项式逆像法
  • 批准号:
    0406450
  • 财政年份:
    2004
  • 资助金额:
    $ 6.35万
  • 项目类别:
    Standard Grant
Harmonic measures, potentials, approximation and polynomials inequalities on general sets
一般集上的调和测度、势、近似和多项式不等式
  • 批准号:
    0097484
  • 财政年份:
    2001
  • 资助金额:
    $ 6.35万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Approximation Theory and Potentials
数学科学:近似理论和势
  • 批准号:
    9415657
  • 财政年份:
    1995
  • 资助金额:
    $ 6.35万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Potential Theoretical Methods in Approximation Theory
数学科学:近似论中潜在的理论方法
  • 批准号:
    9101380
  • 财政年份:
    1991
  • 资助金额:
    $ 6.35万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Special Functions and Orthogonal Expansions with Applications to Nonlinear Dynamics and Quantum Mechanics/Orthogonal Expansions and Potential Theory
数学科学:特殊函数和正交展开及其在非线性动力学和量子力学中的应用/正交展开和势论
  • 批准号:
    9002794
  • 财政年份:
    1990
  • 资助金额:
    $ 6.35万
  • 项目类别:
    Standard Grant

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经典多项式的多面观
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