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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.

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)
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