Polynomials and Exponential Sums: Extremal Problems, Density, Inequalities, and Zeros

多项式和指数和：极值问题、密度、不等式和零点

• 批准号: 0070826
• 负责人: Tamas Erdelyi
• 金额: $ 7.45万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070826&HistoricalAwards=false
• 关键词: Polynomials; Exponential; Sums; Extremal; Problems

Abstract Polynomials pervade mathematics. Virtually every branch of mathematics,from algebraic number theory and algebraic geometry to applied analysis,Fourier analysis, and computer science, has its corpus of theory arising from the study of polynomials. This project intends to study polynomial inequalities and their applications in classical analysis, approximation theory, orthogonal polynomials, and number theory. The proposed research is about polynomials in a general sense, so it includes Chebyshev spaces, Markov spaces, Descartes systems, Muntz polynomials, rational function spaces, as well as polynomials with various constraints such as restricted zeros, integer coefficients, and nonnegative coefficients in various bases. The project continues several years of successful work on a variety of topics and describes various new directions. The polynomial is one of the most basic concepts of mathematics. Throughouthistory people have found problems concerning polynomials especially fascinating. Each of the ``three famous problems'' in the ``Heroic Age'' dealt with zeros of polynomials: squaring the circle, duplicating the cube, and trisecting an angle. Historically, questions relating to polynomials, for example, the solutionof polynomial equations, gave rise to some of the most important problems ofthe day. Besides the natural intellectual interest in them, polynomials, and inparticular extremal problems involving polynomials, arise not only inalmost every field of mathematics, but also in other sciences, especially in engineering. However, it is often the case that polynomials related to a practical problem belong to a restricted class. Depending on the nature of the problem, some additional pieces of information on the polynomial may be known. For filter design, polynomials with coefficients from {-1,0,1} are very useful, because, if they are used, then the filter can be implemented without the use of multiplications; their implementation requires only additions and subtractions. This reduces the cost of implementation. The problem is to find such polynomials which have a ``lowpass'' behavior on the unit circle of the complex plane. Roughly, that is, they approximate 1 in a neighborhood of 1 and approximate 0 in a neighborhood of -1. The project plans to study extremal properties of polynomials with coefficients from {-1,0,1}{0,1}, and {-1,1}. Unimodular polynomials (when the coefficients are complex numbers of modulus 1) are also important in engineering. In some other problems related to physics, in particular electrostatics, information about the distribution of the zeros of the polynomial is known. ``Gaped'' polynomials (polynomialswith a large number of zero coefficients) are used to analyze, for example,the time-optimal boundary controls for the one-dimensional heat equation. The proposal intends to explore further applications of polynomial inequalitiesas well as to offer a systematic study of naturally arising questions regarding polynomials subject to various constraints.

摘要 多项式在数学中无处不在。事实上，数学的每个分支，从代数数论和代数几何到应用分析、傅立叶分析和计算机科学，都有其源自多项式研究的理论语料库。该项目旨在研究多项式不等式及其在经典分析、逼近论、正交多项式和数论中的应用。所提出的研究是关于一般意义上的多项式，因此它包括切比雪夫空间、马尔可夫空间、笛卡尔系统、蒙兹多项式、有理函数空间，以及各种具有限制零点、整数系数和非负系数等各种约束的多项式。基地。 该项目延续了多年来在各种主题上的成功工作，并描述了各种新方向。 多项式是数学最基本的概念之一。纵观历史，人们发现有关多项式的问题尤其令人着迷。 “英雄时代”的“三大著名问题”都涉及多项式的零点：圆的平方、立方体的倍数和角的三等分。从历史上看，与多项式相关的问题，例如多项式方程的解，引起了当今一些最重要的问题。除了对多项式的天然智力兴趣之外，多项式，特别是涉及多项式的极值问题，不仅出现在数学的几乎每个领域，而且出现在其他科学中，特别是在工程学中。然而，通常情况下，与实际问题相关的多项式属于受限类。根据问题的性质，多项式的一些附加信息可能是已知的。对于滤波器设计，系数为 {-1,0,1} 的多项式非常有用，因为如果使用它们，则可以在不使用乘法的情况下实现滤波器；它们的实现只需要增加和减少。这降低了实施成本。问题是找到在复平面的单位圆上具有“低通”行为的多项式。粗略地说，它们在 1 的邻域中近似 1，在 -1 的邻域中近似 0。该项目计划研究系数为 {-1,0,1}{0,1} 和 {-1,1} 的多项式的极值性质。幺模多项式（当系数是模 1 的复数时）在工程中也很重要。在与物理（特别是静电学）相关的一些其他问题中，有关多项式零点分布的信息是已知的。例如，“Gaped”多项式（具有大量零系数的多项式）用于分析一维热方程的时间最优边界控制。该提案旨在探索多项式不等式的进一步应用，并对受各种约束的多项式自然产生的问题进行系统研究。