Mathematical Sciences: Polynomials and Polynomial Inequalities

摘要：DMS-9623156 PI: Erdelyi多项式遍及数学。实际上，数学的每一个分支，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，都有其从多项式研究中产生的理论语料库。本课题旨在研究多项式不等式及其在经典分析、近似理论、正交多项式和数论中的应用。本课题研究的是广义上的多项式，包括切比雪夫空间、马尔可夫空间、笛卡儿系统、蒙兹多项式、有理函数空间，以及各种基上具有限制零、整数系数、非负系数等约束的多项式。该项目延续了几年来在各种主题上的成功工作，并描述了各种新的方向。多项式是数学中最基本的概念之一。纵观历史，人们发现多项式的问题特别令人着迷。“英雄时代”的“三大著名问题”都涉及多项式的零：圆的平方、立方体的复制和角的三切分。从历史上看，与多项式有关的问题，例如多项式方程的解，产生了当今一些最重要的问题。除了对多项式的自然兴趣之外，多项式，特别是涉及多项式的极端问题，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}的多项式是非常有用的，因为如果使用它们，则无需使用乘法即可实现滤波器；它们的实现只需要加法和减法。这降低了实现的成本。问题是在复平面的单位圆上找到具有“低通”行为的多项式。粗略地说，它们在邻域为1的范围内近似1，在邻域为-1的范围内近似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。单模多项式（系数为模1的复数）在工程中也很重要。在其他一些与物理有关的问题中，特别是静电学，关于多项式零点分布的信息是已知的。“间隙”多项式（具有大量零系数的多项式）用于分析，例如，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，以及对受各种约束的多项式自然产生的问题进行系统研究。

ABSTRACT Proposal: DMS-9623156 PI: Erdelyi 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. Polynomials are one of the most basic concepts of mathematics. Throughout history 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 solution of polynomial equations, gave rise to some of the most important problems of the day. Besides the natural intellectual interest in them, polynomials, and in particular extremal problems involving polynomials, arise not only in almost 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 implementati on 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 (with coefficients that 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 (polynomials with 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 inequalities as well as to offer a systematic study of naturally arising questions regarding polynomials subject to various constraints.