an:01874240
Zbl 1012.65025
Rambour, Philippe; Seghier, Abdellatif
Exact and asymptotic inverse of the Toeplitz matrix with polynomial singular symbol
EN
C. R., Math., Acad. Sci. Paris 335, No. 8, 705-710 (2002), erratum 336, No. 5, 399-400 (2003).
00090408
2002
j
65F05 65F40 15A15 15B57 47B35 15A09 60G25
exact inverse matrix; asymptotic inverse matrix; polynomial singular symbol; Toeplitz matrix; Green kernels; differential operators; traces; determinants
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.