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

Abstract

This paper surveys the closely related topics included in the title. Emphasis is given to the parallelism between the approach using (Perron-Caratheodory) continued fractions to solve the trigonometric moment problem, and the alternate development that proceeds from the sequence of moments {ti n }™«>> to the linear functional ft, to the Szego polynomials and their reciprocal and associated polynomials, and to the quadrature formula for fi and the solution of the moment problem. CONTENTS

Sieved para-orthogonal polynomials on the unit circle

“…Such results in a setting based on linear algebra and also without the use of the name para orthogonal appear, even earlier than in [12], in Gragg [13]. However, the name para orthogonal polynomials for S n ðzÞ s n S Ã n ðzÞ, where js n j 1 and S n are OPUC, is due to Jones et al [17]. We may refer to the polynomials ðz wÞP n ðw; zÞ as the CD kernel POPUC.…”

“…Hence, S nþ1 ðzÞ s nþ1 ðwÞS Ã nþ1 ðzÞ is known as a para orthogonal polynomial associated with S nþ1 . From known properties of para orthogonal polynomials (see [17]), S nþ1 ðzÞ s nþ1 ðwÞS Ã nþ1 ðzÞ has n þ 1 simple zeros on the unit circle jzj 1. In particular, w is one of the zeros of S nþ1 ðzÞ s nþ1 ðwÞS Ã nþ1 ðzÞ.…”

A matrix approach to the computation of quadrature formulas on the unit circle

“…On the other hand, the rapidly growing interest on problems on the unit circle, like quadratures, Szegő polynomials and the trigonometric moment problem has suggested to develop a theory of orthogonal Laurent polynomials on the unit circle introduced by Thron in [36], continued in [26], [21], [11] and where the recent contributions of Cantero, Moral and Velázquez in [4], [3] and [6] has meant an important and definitive impulse for the spectral analysis of certain problems on the unit circle. Here, it should be remarked that the theory of orthogonal Laurent polynomials on the unit circle establishes features totally different to the theory on the real line because of the close relation between orthogonal Laurent polynomials and the orthogonal polynomials on the unit circle (see [9]).…”

“…A proof in the ordinary polynomial situation (i.e. p(n) = 0 for all n) is given in [26] and an alternative simpler approach in [15] (see also [1]). A proof based on the techniques introduced in the Chihara's book [7] in the balanced situations p(n) = E n+1 2 and p(n) = E n 2 is given in [9].…”

“…[26]), it will be said that the functional µ is quasi-definite if and only if the principal submatrices of the infinite Toeplitz moment matrix associated with {µ n } ∞ −∞ are nonsingular and positive-definite if the determinants of these matrices are positive. Quasi-definiteness is a necessary and sufficient condition for the existence of a family of orthogonal Laurent polynomials with respect to the linear functional (35) in the sense that there exists a sequence {R n (z)} ∞ n=0 of Laurent polynomials satisfying R n (z) ∈ L n \L n−1 , n = 1, 2, .…”