Toeplitz determinants with piecewise continuous generating function

Abstract

Consequent application of the theory of operator determinants and a special technique of perturbing by trace class operators allow to determine the asymptotic behavior of the Toepiitz determinants $D_n(a)=\det \{a_{j-k}{\}}_{j,k-0}^n(n\to \infty )$, if the generating function $a(t)={\sum }_{k=-\infty }^{\infty }{a}_k{t}^k(|t|=1)$ is piecewise continuous and satisfies some natural conditions of regularity. There holds $D_n(a)\sim G^{n+1}\cdot E\cdot n^{-{\sum }_{r=1}^R{\beta }_r^2}(n\to \infty )$, where ${\beta }_r=\frac{1}{2\pi i}$ log $a(t_r-0)/a(t_r+0)$, |Re ${\beta }_r$| < 1/2 with $t_1,\dots ,t_R$ being the points of discontinuity of $a(t)$; thereby the constants $G$ and $B$ are explicitely given.