Eigenvalues of Hermitian Toeplitz matrices with polynomially increasing entries
The authors consider the first order asymptotics, as \(n\to \infty\), of the extreme eigenvalues of the \(n\times n\) Hermitian Toeplitz matrices whose first rows have the form \([a_0,a_1,\dots ,a_{n-1}]\) where, for some \(\alpha >0\), \(a_k=k^{\alpha}+o(k^{\alpha})\) as \(k\to \infty\). Following an approach introduced by \textit{H. S.\ Widom} [Trans.\ Am.\ Math.\ Soc., 88, 491--522 (1958; Zbl 0101.09202)], they relate the eigenvalues of these matrices to those of an integral operator with kernel \(K(x,y)=|x-y|^{\alpha}\). More details are given for some special cases. In one of these cases, their results provide an alternative proof of a result of W. F.\ Trench.
scientific article; zbMATH DE number 6064844