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

琼斯提出了四个密切相关的研究领域：(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。以下是对拟议研究的简要说明。(1)强矩问题的研究将集中于当已知存在多个解时确定给定问题的所有解。对于某些类型的问题，我们建议证明存在无穷多个阶跃函数形式的解。对于与双对数正则矩序列相关的无限族矩问题，我们提出了利用N evanlinna参数化法得到矩问题所有解的显式表达式。(2)这一领域研究的一个主要目标是确定类似于经典的Jacobi、Hermite和Laguerre的正交多项式的OLP的显式公式。还将为递推公式、罗德里格斯公式、微分方程、求积公式和经典多项式的其他已知性质寻求显式公式。对于(3)和(4)领域，我们的主要目标是研究用于将最近获得的理论结果应用于科学和技术问题的计算方法。正交函数被用作用数学语言描述大量物理现象的积木；例如：(A)地球磁场(用于导航和地球物理研究)，(B)音乐音调，(C)海潮运动，(D)交变电流和(E)由于引力(绕行星和卫星运行)和静电学而产生的力场。有助于形成数学物理(及其工程应用)基础的正交函数的例子有球谐函数和柱谐函数、傅里叶级数和正交多项式。OLPS是由主要研究者和合作者(Thron，Waadland和Njastad)在1980年代的S中引入的一种相对较新的正交函数。最初，它们被用来解决力矩问题。琼斯建议将它们应用到求积公式中，以便有效地计算科学和工程中出现的积分。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、语音治疗和聋人说话等领域中出现的频率分析问题。我们的研究将把Szego多项式方法与其他已知的方法进行比较，以求对Szego方法进行改进并开发其应用。

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.