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Formulas for the inverses of Toeplitz matrices with polynomially singular symbols. (English) Zbl 1069.47027
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 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 Theory 50, No. 1, 43–55 (2004; Zbl 1070.47015)] and the historical references therein):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.

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
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