Toeplitz Determinants with One Fisher–Hartwig Singularity

Abstract

Letcbe a function defined on the unit circle with Fourier coefficients {cn}∞n=−∞. The Fisher–Hartwig conjecture describes the asymptotic behaviour of the determinants of then×nToeplitz matricesDn(c)=det[ci−j]n−1i, j=0for a certain class of functionsc. In this paper we prove this conjecture in the case of functions with one singularity. More precisely, we consider functions of the formc(eiθ)=b(eiθ)tβ(ei(θ−θ1))uα(ei(θ−θ1)).Heretβ(eiθ)=exp(iβ(θ−π)), 0<θ<2π, is a function with a jump discontinuity,uα(eiθ)=(2−2cosθ)αis a function which may have a zero, a pole, or a discontinuity of oscillating type, andbis a sufficiently smooth nonvanishing function with winding number equal to zero. The only restriction we impose on the parameters is that 2αis required not to be a negative integer. In the case where Reα⩽−1/2, i.e., where the corresponding functioncis not integrable, we identifycin an appropriate way with a distribution.