Toeplitz minors (Q1604561)

Let \(T_{n-1} (f)\) denote the \(n\times n\) Toeplitz matrix generated by the Fourier coefficients of the function \(f\) on the unit circle \(\mathbf{T}\) and let \(D_{n-1} (f)=\det T_{n-1} (f)\). The well known strong Szegö limit theorem asserts that if \(\sigma (t)=\exp \left(\sum_{n=-\infty}^\infty c_nt^n\right)\) then \(D_{n-1}(\sigma)\sim \exp\left(nc_0+\sum_{k=1}^\infty kc_kc_{-k}\right)\) under some assumptions on \(\sigma\). The authors give a new proof of this theorem using symmetric function theory. The application to asymptotics for Toeplitz minors is also given.