Formulas for the inverses of Toeplitz matrices with polynomially singular symbols
Let \(T\) be the unit circle and \(T_N(h)\) be the \((N+1)\times (N+1)\) Toeplitz matrix \((\widehat h(k-\ell))_{k,\ell}^N\), where \(\widehat h(k)\) are the Fourier coefficients of the function \(h(t)=|1-t|^{2p}f_1(t), t\in T, p\in \mathbb N\). The authors study the asymptotic behavior of the \(k,\ell\) entry \([T_N^{-1} (h)]_{k,\ell}\) as \(N\to\infty\). Denote \(A_1(T)=\{a\in L^1(T):\sum_{k\in \mathbb Z}(|k|+1)|{\widehat a}(k)|< \infty\}\) and let \(f_1>0, f_1\in A_1(T).\) It is established that \[ [T_N^{-1} (h)]_{[Nx]+1,[Ny]+1}= \frac{1}{f_1(1)}G_p(x,y)N^{2p-1}+o(N^{2p-1}) \] as \(N\to\infty\) uniformly with respect to \(x\) and \(y\) in \([0,1]\), where \(G_p(x,y)\) can be identified, with the Green kernel associated to the differential operator \((-1)^p\frac{d^{2p}}{dx^{2p}}\) with the boundary conditions \(f_1^{(0)}(0)=\dots=f_1^{(p-1)}(0)=0, f_1^{(0)}(1)=\dots =f_1^{(p-1)}(1)=0\) . The authors remark that the expression for \(G_p(x,y)\) may also be given by a formula due \textit{A. Böttcher} (see [Integral Equations Oper. Theory 50, No. 1, 43--55 (2004; Zbl 1070.47015)] and the historical references therein): \[ G_p(x,y)=\frac{x^py^p}{[(p-1)!]^2} \int_y^1\frac{(t-x)^{p-1}(t-y)^{p-1}}{t^{2p}} \,dt. \] and \(G_p(0,0)=0\) . In the case \(f_1=g_1\overline{g_1}\), where \(g_1,g_1^{-1}\in H^\infty\), \(g_1(0)>0\), the following asymptotic formula for \(0<x\leq 1\) is valid: \[ [T_N^{-1} (h)]_{[Nx]+1,1}= \frac{1}{g_1(0)c_p(g_1(1))} \frac{x^{p-1}(1-x)^p}{(p-1)!}N^{p-1}+o(N^{p-1}) \] as \(N\to\infty\) uniformly in \([\delta, 1]\) for all \(\delta\), \(0<\delta<1\). Here \(c_p(z)=z\) if \(p\) is odd and \(c_p(z)=\overline z\) if \(p\) is even.
scientific article; zbMATH DE number 2113887