Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle. (English) Zbl 0637.30035

Jones, William B.; Njåstad, Olav; Thron, W. J.

Bull. Lond. Math. Soc. 21, No. 2, 113-152 (1989).

Szegö polynomials, the trigonometric moment problem and quadrature on the unit circle have received much recent attention as a result of their applications in the rapidly growing field of digital signal processing, in operator theory, and in probability theory. Many of the known results in this area are scattered in a large number of papers (some not easily accessible) dealing with seemingly different topics. Some of the results are known mainly as “folklore” to the workers in the field. Moreover, the original versions are not always presented with the maximum degree of generality. Our purose in this article is to give a unified, concise, easily accessible development of these topics, exploring the limits of validity of various results, and giving proofs with the degree of generality which can be attained without excessive effort.
The development we present gives a central role to continued fractions and hence to three-term recurrence relations. It shows the analogy between the theories of polynomials orthogonal on the unit circle and those orthogonal on the real line. It emphasizes the parallelism that exists between two approaches to the subject: one of these is based on continued fractions and their modified approximants; the other approach proceeds from the sequence of moments \(\{\mu _ n\}^{\infty}_{- \infty}\) to the linear functional \(\mu\), to the Szegö polynomials their reciprocal and associated polynomials, and finally to a quadrature formula for \(\mu\) that enables us to solve the moment problem. The latter path is analogous to one introduced by M. Riesz for the Hamburger moment problem.

Reviewer: W.B.Jones

MSC:

30E05	Moment problems and interpolation problems in the complex plane
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
41A21	Padé approximation
41A55	Approximate quadratures

Keywords:

Szegö polynomials; moment problem; quadrature; continued fractions