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.

抽象多项式遍布数学。实际上，数学的每个分支，从代数数理论和代数几何形状到应用分析，傅立叶分析和计算机科学的每个分支，都源于多项式研究的理论语料库。该项目旨在研究多项式不平等现象及其在经典分析，近似理论，正交多项式和数量理论中的应用。拟议的研究是关于一般意义上的多项式的，因此它包括Chebyshev空间，马尔可夫空间，笛卡尔系统，Muntz多项式，有理功能空间以及具有限制性零的多项式，例如零底，整数系数，非底基系数，以及各个基碱基的非负系数。 该项目在各种主题上持续了数年的成功工作，并描述了各种新方向。 多项式是数学最基本的概念之一。通过inthistory的人发现有关多项式的问题，特别令人着迷。 ``英雄时代''中的``三个著名问题''都涉及多项式的零：将圆圈平方，重复多维数据集并进行三角角度。从历史上看，与多项式有关的问题，例如多项式方程式的解决方案引起了当天最重要的问题。除了对它们的自然智力兴趣，多项式和涉及多项式的内部极端问题外，不仅出现在数学的每个领域，而且还出现在其他科学领域，尤其是在工程学中。但是，通常情况下，与实际问题相关的多项式属于限制类别。根据问题的性质，可以知道有关多项式的一些其他信息。对于滤波器设计，具有来自{-1,0,1}系数的多项式非常有用，因为如果使用它们，则可以在不使用乘法的情况下实现过滤器；他们的实施仅需要增加和减法。这降低了实施成本。问题是要找到在复杂平面的单位圆上具有``低通''行为的多项式。大致，也就是说，它们在1个附近的1中大约为1，在-1附近大约为0。该项目计划以{-1,0,1} {0,1}和{-1,1}的系数研究多项式的极端特性。单型多项式（当系数为模量的复数时）在工程中也很重要。在与物理学有关的其他一些问题中，已知有关多项式零的分布的信息。 ``GAPAPED''多项式（多项式设备都有大量的零系数）用于分析，例如，一维热方程的时间 - 最佳边界控制。该提案旨在探讨多项式不等式的进一步应用，以及对自然产生的有关多项式问题的自然问题进行系统的研究。