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

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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.