The Tridiagonal Approach to Szegö’s Orthogonal Polynomials, Toeplitz Linear Systems, and Related Interpolation Problems

The basic topics of the paper are the three-term recurrence relation $x_{k + 1} (z) = (\alpha _k + \bar \alpha _k z)x_k (z) - zx_{k - 1} (z)$ and the associated tridiagonal matrix. This relation, which underlies the Bistritz stability test, can be used as a starting point for a novel approach to the trigonometric moment problem and its relatives. In particular, the "tridiagonal approach" is shown to provide a new solution method for the classical Caratheodory-Fejer and Nevanlinna-Pick interpolation problems. The results include some Levinson-type and Schur-type algorithms, of reduced complexity, for computing reflection coefficients associated with nonnegative definite Hermitian Toeplitz matrices.