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

The Fourier extension method and discrete orthogonal polynomials on an arc of the circle