Formal orthogonal polynomials and Hankel/Toeplitz duality
For classical polynomials orthogonal with respect to a positive measure supported on the real line, the moment matrix has a Hankel structure and is positive definite. The polynomials satisfy a three-term recurrence relation and a Christoffel-Darboux type formula. When, as in the Szegö theory, the measure is supported on the complex unit circle, then the moment matrix has a Toeplitz structure, and is strongly regular and positive definite. These Szegö polynomials also satisfy a three-term recurrence relation, but also a coupled recurrence relation. In this interesting paper, the authors study the generalization for formal polynomials orthogonal with respect to an arbitrary moment matrix and consider arbitrary Hankel and Toeplitz matrices as special cases. They show how much of the classical theory of Hankel and Toeplitz matrices can be generalized. A new relation between Schur complements and the Euclidean algorithm is obtained. The relations with Padé approximation, continued fractions and Krylov subspace methods in linear algebra are also outlined.
scientific article; zbMATH DE number 826914
scientific article