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Tridiagonal approach to the algebraic environment of Toeplitz matrices. II: Zero and eigenvalue problems. (English) Zbl 0753.15015

[For part I see ibid. 12, No. 2, 220-238 (1991; Zbl 0728.65020).]
The authors continue their investigation of families of symmetric and antisymmetric predictor polynomials in connection with Hermitian Toeplitz matrices and associated reflection coefficients.
These polynomials satisfy a three-term recurrence relation, and analogies with the classical theory of orthogonal polynomials. The important problem of computing the zeros of the highest-degree symmetric polynomial is shown to be equivalent to the eigenvalue problem for a unitary Hessenberg matrix.
For real data (reflection coefficients), the problem can be further transformed to the eigenvalue problem for a symmetric tridiagonal matrix.

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
15A18 Eigenvalues, singular values, and eigenvectors
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
60G25 Prediction theory (aspects of stochastic processes)

Citations:

Zbl 0728.65020
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