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

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)
Full Text: DOI
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