Exact and asymptotic inverse of the Toeplitz matrix with polynomial singular symbol
Philippe Rambour, Abdellatif Seghier · Comptes Rendus Mathématique · 2002
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 non negative regular function and P a polynomial with all its zeros on 𝕋 . In a second part we give an asymptotic expansion for symbols (1−cos θ ) p f when f is always a nonnegative regular function. These formulas use Green kernels associated to differential operators of order 2 p . Finally, we propose some applications to the computation of traces and determinants.