The Extended Fisher-hartwig Conjecture for Symbols with Multiple Jump Discontinuities

The asymptotic expansions of Toeplitz determinants of certain symbols with multiple jump discontinuities are shown to satisfy a revised version of the conjecture of Fisher and Hartwig. The Toeplitz matrix T n [φ] is said to be generated by the function φ if T n [φ] = (φ i−j), i, j = 0,. .. , n − 1. where φ n = 1 2π 2π 0 φ(θ)e −inθ dθ is the nth Fourier coefficient of φ. Define the determinant The Fisher-Hartwig Conjecture [8] concerns the asymptotic behavior of the determinants of Toeplitz matrices for a certain class of singular symbols. These symbols are of the form φ(θ) = b(θ) R r=1 t βr (θ − θ r)u αr (θ − θ r) (2)