Determinants and Eigenvalues

We now study the behavior of the determinants $${D_n}(a): = \det {T_n}(a): = \det \left( {\begin{array}{*{20}{c}} {{a_0}}&{{a_{ - 1}}}& \cdots &{{a_{ - (n - 1)}}} \\ {{a_1}}&{{a_0}}& \cdots &{{a_{ - (n - 2)}}} \\ \vdots & \vdots & \ddots & \vdots \\ {{a_{n - 1}}}&{{a_{n - 2}}}& \cdots &{{a_0}} \end{array}} \right)$$ as n goes to infinity. The strong Szegö limit theorem says that, after appropriate normalization, the determinants D_n(a) approach a nonzero limit provided a is sufficiently smooth and T(a) is invertible. Before stating and proving this theorem, we need a few more auxiliary facts.