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Zeitschrift für Analysis und ihre Anwendungen


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Volume 1, Issue 2, 1982, pp. 23–39
DOI: 10.4171/ZAA/12

Published online: 1982-04-30

Toeplitz determinants with piecewise continuous generating function

Albrecht Böttcher[1]

(1) Technische Universität Chemnitz, Germany

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 $D_n(a) = \mathrm {det} \: \{a_{j-k}\}^n_{j,k-0} (n \to \infty)$, if the generating function $a(t) = \sum^{\infty}_{k=-\infty}a_k t^k (|t| = 1)$ is piecewise continuous and satisfies some natural conditions of regularity. There holds $D_n(a) \sim G^{n+1} \cdot E \cdot n^{-\sum^R_{r=1} \beta_r^2} (n \to \infty)$, 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.

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Böttcher Albrecht: Toeplitz determinants with piecewise continuous generating function. Z. Anal. Anwend. 1 (1982), 23-39. doi: 10.4171/ZAA/12