A Riemann-Hilbert approach to some theorems on Toeplitz operators and orthogonal polynomials

In this paper, the authors show how to use Riemann-Hilbert techniques to prove various results, some old, some new, in the theory of Toeplitz operators and orthogonal polynomials on the unit circle (OPUCs). There are four main results: the first concerns the approximation of the inverse of a Toeplitz operator by the inverses of its finite truncations. The second concerns a new proof of the 'hard' part of Baxter's theorem, and the third concerns the Born approximation for a scattering problem on the lattice Z₊. The fourth and final result concerns a basic proposition of Golinskii-Ibragimov arising in their analysis of the Strong Szegö Limit Theorem.