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

摘要Jones提出了四个密切相关的研究方向：(1)强矩理论，(2)正交洛朗多项式序列，(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描述。(1)当已知存在多个解时，对强矩问题的研究将集中于确定给定问题的所有解。对于某类问题，我们提出了以阶跃函数形式存在无穷多个解的证明。对于与对数正态矩双序列相关的无穷一族矩问题，我们利用Nevanlinna参数化，给出了该矩问题所有解的显式表达式。(2)该领域研究的主要目标是确定OLPS的显式公式，这些公式类似于Jacobi， Hermite和Laguerre的经典正交多项式。还将寻求显式公式的递归公式，罗德里格斯公式，微分方程，正交公式和其他已知的经典多项式的性质。对于领域(3)和(4)，我们的主要目标是研究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。用正交函数作为构建块，用数学语言描述大量的物理现象；例如：(a)地球磁场（用于导航和地球物理研究），(b)音乐音调，(c)海洋潮汐运动，(d)交流电流和(e)由于引力（绕轨道运行的行星和卫星）和静电而产生的力场。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面和圆柱谐波、傅立叶级数和正交多项式。OLPS是由首席研究员和合作者（Thron， Waadeland和Njastad）在20世纪80年代引入的相对较新的正交函数。最初它们是用来解决强矩问题的。Jones建议将它们应用于科学和工程中出现的积分高效计算的正交公式。Szego多项式适用于通信（广播、电视、电话信号的数字传输）、雷达、语音学、语音处理、语言治疗、聋人说话教学等领域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方法，寻求改进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.