Measures associated with Toeplitz matrices generated by the Laurent expansion of rational functions

Abstract:

Let ${T_n}(a) = ({a_{i - j}})_{i,j = 0}^n$ be the finite Toeplitz matrices generated by the Laurent expansion of an arbitrary rational function, and let ${\sigma _n} = \{ {\lambda _{n0}}, \ldots ,{\lambda _{nn}}\}$ be the corresponding sets of eigenvalues of ${T_n}(f)$. Define a sequence of measures ${\alpha _n},{\alpha _n}(E) = {(n + 1)^{ - 1}}{\Sigma _{{\lambda _{ni}} \in E}}1,{\lambda _{ni}} \in {\sigma _n}$, and $E$ a set in the $\lambda$-plane. It is shown that the weak limit $\alpha$ of the measures ${\alpha _n}$ is unique and possesses at most two atoms, and the function $f$ which give rise to atoms are identified.