The constants in the asymptotic formulas by Rambour and Seghier for inverses of Toeplitz matrices. (English) Zbl 1070.47015

Böttcher, A.

Integral Equations Oper. Theory 50, No. 1, 43-55 (2004).

Let \(T\) be the unit circle and \(a(t)=|1-\tau|^{2p}f(\tau), \tau\in T\), where \(p\in\mathbb{N}\) and \(f\) is a positive continuous function such that \(\sum_{n\in \mathbb{Z}} |n||f_n|<\infty\), and \(f_n\) are the Fourier coefficients. P. Rambour and A. Seghier [C. R. Acad. Sci., Paris 335, No. 8, 705–710 (2002; Zbl 1012.65025); erratum: ibid. 336, 399–400 (2003); Integral Equations Oper. Theory 50, No. 1, 83–114 (2004; Zbl 1069.47027)] discussed the asymptotic behavior of the \(j,k\) entry \([T_n^{-1} (a)]_{j,k}\) as \(n\to\infty\) of the Toeplitz \((n+1)\times (n+1)\) matrix \(T_n^{-1} (a)=(T_n (a))^{-1},\;T_n (a)=(a_{j-k})_{j,k=0}^n\). They established that \[ [T_n^{-1} (a)]_{[nx],[ny]}= \frac{1}{f(1)}G_p(x,y)n^{2p-1}+o(n^{2p-1}), \] as \(n\to\infty\) uniformly with respect to \(x\) and \(y\) in \([0,1]\). They also remarked that \(G_1(x,y)=x(1-y), G_2(x,y)=\frac 16 x^2(1-y)^2 (3y-x-2xy).\)
In the present paper, the author gives an elegant method for the calculation of \(G_p(x,y)\) for any \(p\in \mathbb{N}\).
Theorem. For \(0\leq x\leq 1\) and \(y\geq \max (x,1-x)\), \[ G_p(x,y)=\frac{x^py^p}{[(p-1)!]^2}\int_y^1\frac{(t-x)^{p-1}(t-y)^{p-1}}{t^{2p}}dt. \]
The asymptotic behavior of the trace and sum of all entries are also considered.

Reviewer: Nikolai K. Karapetyants (Rostov-na-Donu)

MSC:

47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
15A09	Theory of matrix inversion and generalized inverses
34B27	Green’s functions for ordinary differential equations
47N20	Applications of operator theory to differential and integral equations
47N30	Applications of operator theory in probability theory and statistics

Keywords:

Toeplitz matrix; singular symbol; inverse matrix; Green’s function