Toeplitz matrices with an exponential growth of entries and the first Szegő limit theorem
Let \(S(n)=\{s_{j-k}\}_{k,j=1}^n, n\leq\infty\), be a block Toeplitz matrix. The author studies such matrices with an exponential growth of entries and its connection with the first Szegő limit theorem. With Toeplitz matrix are connected two analytic at zero functions \(\varphi (\lambda)=\frac{s_0}{2}+\sum_{p=1}^\infty s_{-p}\lambda^p\) and \(\psi (\mu)=\frac{s_0}{2}+\sum_{p=1}^\infty s_{p}\mu^p\). The poles of \(\varphi\) and \(\psi\) inside unit circle allow an exponential growth of \(\|s_{\pm p}\|\). The main idea is based on replacing \(S(\infty)\) by a matrix \(S(\infty)=\{s_{j-k}\}_{k,j=1}^\infty\) with better behaviour of entries and using asymptotic \(\det S(r), r\to\infty\) instead of \(\det S(r)\). By this way a Szegő-type limit theoerm is also obtained.
scientific article; zbMATH DE number 1437361