Remarks on extreme eigenvalues of Toeplitz matrices
Let f be a nonnegative integrable function on \([-\pi,\pi]\), \(m=0\) its essential infimum matrix associated with f and \(\lambda_{1,n}\) its smallest eigenvalue. It follows from a theorem of Szegö that \(\lambda_{1,n}\to 0\) as \(n\to \infty\). The author discussed the rate of this convergence and obtained a lower bound for \(\lambda_{1,n}\) without any smoothness conditions. The main results are the following. Theorem 3.2 Let \(E=\{\theta:\) \(f(\theta)>0\}\) and \(\tau^*(E)\) be its outer measure, then \(\lim\sqrt{\lambda_{1,n}}\leq \tau^*(E)\). Therefore, if \(\tau^*(E)<1\), then \(\lambda_{1,n}\) decrease to zero at least exponentially as \(n\to \infty\). Theorem 3.6 If \(f^{-1}\in L\), then for all \(n\geq 1\), we have \(\lambda_{1,n}\geq 1/\pi^ 2C(n+1)^ 2\) where C is a constant depending on f and \(C=\int f^{-1}d\theta /2\pi\).
extreme eigenvalues of Toeplitz matrices
scientific article; zbMATH DE number 4040626