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Asymptotic behavior of variable-coefficient Toeplitz determinants. (English) Zbl 0990.47024
Let $$\sigma$$ be a continuous function on $$[0,1]\times {\mathbb T}$$, where $$\mathbb T$$ is the unit circle. Denote by $$op_n \sigma$$ the $$(n+1)\times (n+1)$$ matrix whose $$(j,k)$$-entris are given by $${1 \over 2\pi} \int_0^{2\pi} \sigma ({j\over n}, e^{-i(j-k)\theta}) d\theta$$. These matrices can be thought of as variable-coefficient Toeplitz matrices or as the discrete analogues of pseudodifferential operators. For functions $$\sigma$$ with sufficiently smooth logarithms the authors establish the following asymptotic formula $$\det [op_n \sigma] \sim G[\sigma]^{(n+1)} E[\sigma]$$, as $$n \rightarrow \infty$$. The constants $$G[\sigma]$$ and $$E[\sigma]$$ are explicitly determined.

##### MSC:
 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 15A15 Determinants, permanents, traces, other special matrix functions
##### Keywords:
Toeplitz determinant; Szegő limit theorem
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