Moment Theory, Orthogonal Polynomials, Quadrature, and Continued Fractions Associated with the unit Circle

1989/03/01 by William B. Jones, Olav Njåstad, W. J. Thron · 18 citations
Mathematics · Physics and Astronomy · #Mathematical functions and polynomials #Fractional Differential Equations Solutions #Electromagnetic Scattering and Analysis

paper · doi:10.1112/blms/21.2.113

Abstract

This paper surveys the closely related topics included in the title. Emphasis is given to the parallelism between the approach using (Perron–Carathéodory) continued fractions to solve the trigonometric moment problem, and the alternate development that proceeds from the sequence of moments μn−∞∞, to the linear functional μ, to the Szegö polynomials and their reciprocal and associated polynomials, and to the quadrature formula for μ and the solution of the moment problem.

Cited by

• The Statistical Distribution of the Zeros of Random Paraorthogonal Polynomials on the Unit Circle
• Poisson Brackets of Orthogonal Polynomials
• Quantum Krylov Algorithm for Szegö Quadrature
• Weak convergence of CD kernels and applications
• Refined interlacing properties for zeros of paraorthogonal polynomials on the unit circle
• Bi-orthogonal Polynomials on the Unit Circle, regular semi-classical Weights and Integrable Systems
• Measures and semi-orthogonal functions on the unit circle
• Wall rational functions and Khrushchev's formula for orthogonal rational functions
• A matrix approach to the computation of quadrature formulas on the unit circle
• Rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle
• Orthogonal polynomials and Möbius transformations
• Orthogonal Polynomials related to g-fractions with missing terms
• Fine Structure of the Zeros of Orthogonal Polynomials, I. A Tale of Two Pictures
• R_II type recurrence, generalized eigenvalue problem and orthogonal polynomials on the unit circle
• Differential equations for a class of semiclassical orthogonal polynomials on the unit circle
• Positive trigonometric Quadrature Formulas and quadrature on the unit\n circle
• Markov theorem for weight functions on the unit circle
• Measures on the unit circle and unitary truncations of unitary operators

Related

• Continued fractions and orthogonal polynomials in several variables
• Ramanujan's trigonometric sums and para-orthogonal polynomials on the unit circle
• On the trigonometric moment problem
• Matrix measures on the unit circle, moment spaces, orthogonal polynomials and the Geronimus relations
• Constant coefficient Laurent biorthogonal polynomials, Riordan arrays\n and moment sequences