Asymptotic Behavior of Variable-Coefficient Toeplitz Determinants

For a continuous function s\sigma defined on 0,1]×\mathbbT0,1]\times\mathbb{T}, let \ops\op\sigma stand for the (n+1)×(n+1)(n+1)\times(n+1) matrix whose (j,k)(j,k)-entries are equal to \frac1 2pò₀^2p s( \fracjn,e^iq) e^-i(j-k)q  dq,        j,k = 0,1,...,n . \displaystyle \frac{1} {2\pi}\int_0^{2\pi} \sigma \left( \frac{j}{n},e^{i\theta}\right) e^{-i(j-k)\theta} \,d\theta, \qquad j,k =0,1,\dots,n~. These matrices can be thought of as variable-coefficient Toeplitz matrices or as the discrete analogue of pseudodifferential operators. Under the assumption that the function s\sigma possesses a logarithm which is sufficiently smooth on 0,1]×\mathbbT0,1]\times\mathbb{T}, we prove that the asymptotics of the determinants of \ops\op\sigma are given by det\ops] ~ Gs]⁽ⁿ⁺¹⁾Es]     \text as   n?￥ , \det \left\op\sigma\right] \sim G\sigma]^{(n+1)}E\sigma] \quad \text{ as \ } n\to\infty~, where Gs]G\sigma] and Es]E\sigma] are explicitly determined constants. This formula is a generalization of the Szegö Limit Theorem. In comparison with the classical theory of Toeplitz determinants some new features appear.