  The Fourier extension method, also known as the Fourier continuation method,
is a method for approximating non-periodic functions on an interval using
truncated Fourier series with period larger than the interval on which the
function is defined. When the function being approximated is known at only
finitely many points, the approximation is constructed as a projection based on
this discrete set of points. In this paper we address the issue of estimating
the absolute error in the approximation. The error can be expressed in terms of
a system of discrete orthogonal polynomials on an arc of the unit circle, and
these polynomials are then evaluated asymptotically using Riemann--Hilbert
methods.
