Toeplitz and Wiener-Hopf determinants with piecewise continuous symbols

Abstract

The asymptotics for determinants of Toeplitz and Wiener-Hopf operators with piecewise continuous symbols are obtained in this paper. If W α ( σ) is the Wiener-Hopf operator defined on L 2(0, α) with piecewise continuous symbol σ having a finite number of discontinuities at ξ r , then under appropriate conditions it is shown that det W α ( σ) ~ G( σ) α α Σλ r 2 K( σ), where G(σ) = exp(log σ) ( ̂ 0), λ r = ( 1 2π ) log[ σ(ξ r +) σ(ξ r −) ] and K(σ) is a completely determined constant. An analogous result is obtained for Toeplitz operators. The main point of the paper is to obtain a result in the Wiener-Hopf case since the Toeplitz case had been treated earlier. In the Toeplitz case it was discovered that one could obtain asymptotics fairly easily for symbols with several singularities if, for each singularity one could find a single example of a symbol with a singularity of that kind whose associated asymptotics were known. Fortunately in the Toeplitz case such asymptotics were known. The difficulty in the Wiener-Hopf case is that there was not a single singular case where the determinant was explicitly known. This problem was overcome by using the fact that Wiener-Hopf determinants when discretized become Toeplitz determinants whose entries depend on the size of the matrix. No theorem on Toeplitz matrices can be applied directly but these theorems are modified to obtain the desired results.