Universal bounds on the convergence rate of extreme Toeplitz eigenvalues. (Q1874679)

The paper under review is devoted to the study of the asymptotics of the smallest eigenvalue \(\lambda_{\min}(T_n(f))\) of a sequence of \(k\)-level Toeplitz matrices \(T_n(f)\) generated by a real-valued function \(f\in L^1({\mathbb T}^k)\) such that \(\text{ess\,inf} f=0\) and \(\text{ess\,sup} f>0\). The author proves that \(\lambda_{\min}(T_n(f))\geq\exp(-c(n_1+\dots+n_k))\) for every multi-index \((n_1,\dots,n_k)\) and a universal constant \(c\). This was conjectured by \textit{S. Serra Capizzano} [Numer. Funct. Anal. Optimization 21, No. 1--2, 255--261 (2000; Zbl 0955.15003)]. Furthermore, the author investigates Toeplitz matrices \(T_n(\mu)\) generated by positive measures \(\mu\) not necessarily absolutely continuous with respect to the Lebesgue measure. He proves the following nice results: (1) Every \(T_n(\mu)\) is strictly positive definite except when \(\mu\) is a finite linear combination of Dirac masses. (2) If \(\mu\) is absolutely continuous with respect to the Lebesgue measure and not identically zero, then \(\lambda_{\min}T_n(\mu)\geq\exp(-cn)\). (3) If \(\mu\) is not absolutely continuous with respect to the Lebesgue measure, then no a priori bound can be given. (4) If no lower bound like \(\lambda_{\min}T_n(\mu)\geq\exp(-cn)\) holds true, then \(\mu\) is necessarily concentrated on a set of zero Lebesgue measure.