Orthogonal Laurent Polynomials, Moment Theory, Pade Approximants and Continued Fractions

琼斯 摘要 琼斯建议在四个密切相关的领域进行研究：（1）强 矩理论，（2）正交洛朗多项式序列（OLPS）， (3)Szego多项式的应用;（4）特殊函数的计算。 以下是拟议研究的简要说明。 （1）强矩问题的研究将集中在确定 一个给定问题的所有解，当有多个解时， 已知存在。 对于某些类别的问题，我们建议显示 存在无穷多个阶跃函数形式的解。 对于与之相关的矩问题的无限族， 双序列的对数正态矩，我们建议获得明确的 表达式的所有解决方案的时刻问题，通过使用 Nevanlinna参数化。 (2)这方面研究的主要目标是 面积是确定OLPS的显式公式， 经典正交多项式的类似物 还有拉盖尔还将寻求显式公式进行递推 公式，Rodrigues公式，微分方程，求积 经典多项式的公式和其他已知性质。 对于区域（3）和（4），我们的主要目标是研究计算 方法用于应用最近获得的理论结果， 科学技术问题。 正交函数被用作构建块来描述 数学语言描述了大量的物理现象; 例如：（a）地球磁场 (for导航和 地球物理研究），（B）音乐音调，（c）运动 （d）交变电流， (e)引力场（轨道行星 和卫星）和静电学。 正交函数的例子 这有助于形成数学物理（及其工程）的基础， 应用）是球面和圆柱谐波，傅立叶级数 和正交多项式 OLPS是相对新的正交 1980年代由首席调查员引入的职能， 合作者（Thron，Waadeland和Njastad）。 原来他们是 用于解决强矩问题。 琼斯提出， 求积公式的应用 出现在科学和工程中的积分。 Szego多项式适用于通信（无线电、电视和电话信号的数字传输）、雷达、语音学、语音处理、语音治疗和教聋人说话等领域中出现的频率分析问题。 我们的研究将比较Szego多项式方法与其他已知的方法，寻求完善Szego方法，并开发其应用。

Jones Abstract Jones proposes research in four closely related areas: (1) Strong Moment Theory, (2) Orthogonal Laurent Polynomial Sequences (OLPS's), (3) Applications of Szego Polynomials, and (4) Computation of Special Functions. Following are brief descriptions of the proposed research. (1)Research on strong moment problems will be focused on determining all solutions of a given problem when more than one solution is known to exist. For certain classes of problems we propose to show that there are infinitely many solutions in the form of step-functions. For the infinite family of moment problems associated with bi-sequences of log-normal moments, we propose to obtain explicit expressions for all solutions of the moment problems, by using the Nevanlinna parametrization. (2) A primary goal for research in this area is the determination of explicit formulas for OLPS's that are analogues of the classical orthogonal polynomials of Jacobi, Hermite and Laguerre. Explicit formulas will also be sought for recurrence formulas, Rodrigues' formulas, differential equations, quadrature formulas and other known properties of the classical polynomials. For areas (3) and (4) our primary goal is to investigate computational methods for use in applying recently obtained Theoretical results to problems of science and technology. Orthogonal functions are used as building blocks to describe in mathematical language a large number of physical phenomena; for example: (a) the magnetic field of the earth (for navigation and geophysical research), (b) musical tones, (c) motion of ocean tides, (d) alternating electrical currents and (e) force fields due to gravitational attraction (orbiting planets and satellites) and electrostatics. Examples of orthogonal functions that help form the basis of mathematical physics (and its engineering applications) are spherical and cylindrical harmonics, Fourier series and orthogonal polynomials. OLPS's are r elatively new orthogonal functions introduced in the 1980's by the principal investigator and collaborators (Thron, Waadeland and Njastad). Originally they were used to solve strong moment problems. Jones proposes their application to quadrature formulas for efficient computation of integrals that arise in science and engineering. Szego polynomials are applicable to frequency analysis problems that arise in such fields as communication (digital transmission of radio, television and telephone signals), radar, phonetics, speech processing, speech therapy, and teaching the deaf to speak. Our research will compare Szego polynomial methods with other known methods, seeking to refine the Szego method and exploit its application.