Generalized two-term recurrences and fast algorithms for Hermitian Toeplitz matrices

This work extends, unifies and generalizes the relationships between classical polynomial and matrix theory as it relates to the Levinson recurrence for a Hermitian Toeplitz operator. Several new classes of computationally efficient algorithms are presented for the Levinson recurrence on polynomial spaces satisfying certain symmetry properties. This approach also leads to new algorithms for solving systems of linear equations when the coefficient matrix is Hermitian Toeplitz. These algorithms lead to significant improvements in the computational complexity as compared to the previously best known recursive algorithms. They also provide further insight into the mathematical properties of the structurally rich Toeplitz matrices.