Exact and asymptotic inverse of the Toeplitz matrix with polynomial singular symbol. (English. Abridged French version) Zbl 1012.65025

Rambour, Philippe; Seghier, Abdellatif

C. R., Math., Acad. Sci. Paris 335, No. 8, 705-710 (2002), erratum 336, No. 5, 399-400 (2003).

Summary: From a previous work and an application of predictive polynomials we obtain two types of results. In a first part exact entries of the Toeplitz matrix are computed in the case where the symbol is \(|P|^2 f\) and where \(f\) is a nonnegative regular function and \(P\) a polynomial with all its zeros on \(\mathbb{T}\). In a second part we give an asymptotic expansion for symbols \((1- \cos\theta)^p f\) when \(f\) is always a nonnegative regular function. These formulas use Green kernels associated to differential operators of order \(2p\). Finally, we propose some applications to the computation of traces and determinants.

MSC:

65F05	Direct numerical methods for linear systems and matrix inversion
65F40	Numerical computation of determinants
15A15	Determinants, permanents, traces, other special matrix functions
15B57	Hermitian, skew-Hermitian, and related matrices
47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
15A09	Theory of matrix inversion and generalized inverses
60G25	Prediction theory (aspects of stochastic processes)

Keywords:

exact inverse matrix; asymptotic inverse matrix; polynomial singular symbol; Toeplitz matrix; Green kernels; differential operators; traces; determinants

References:

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