Mathematical Sciences: Polynomials and Polynomial Inequalities

摘要提案：DMS-9623156 PI：Erdelyi 多项式在数学中很普遍。事实上，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，都有其理论基础来自于对多项式的研究。本计画主要研究多项式不等式及其在古典分析、逼近论、正交多项式与数论中的应用。所提出的研究是关于一般意义上的多项式，因此它包括Chebyshev空间，Markov空间，Descartes系统，Muntz多项式，有理函数空间，以及各种约束的多项式，如限制零点，整数系数和各种基上的非负系数。 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，人们发现有关多项式的问题特别令人着迷。在“英雄时代”，“三大著名问题”中的每一个都涉及多项式的零点：圆的平方，立方体的复制和三等分。 角度从历史上看，与多项式有关的问题，例如多项式方程的解，引起了当时一些最重要的问题。除了自然的智力兴趣，他们，多项式，特别是极值问题涉及多项式，出现不仅在几乎每一个领域的数学，而且在其他科学，特别是在工程。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，多项式上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，则滤波器可以在不使用乘法的情况下实现;它们的实现仅需要加法和减法。这降低了实施成本。问题是要找到这样的多项式，有一个“低通”的行为单位圆的复平面。粗略地说，它们在1的邻域中近似为1，在-1的邻域中近似为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。在其他一些与物理学相关的问题中，特别是静电学，关于多项式的零点分布的信息是已知的。例如，“间隙”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，以及提供一个系统的研究自然产生的问题，多项式受到各种约束。

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.