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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.

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
15A15 Determinants, permanents, traces, other special matrix functions
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