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

## TL;DR

This paper analyzes the Fourier extension method for approximating functions on an interval, focusing on error estimation using discrete orthogonal polynomials on a circle arc and asymptotic evaluation via Riemann-Hilbert techniques.

## Contribution

It introduces a new approach to estimate approximation error by linking it to discrete orthogonal polynomials and applying Riemann-Hilbert analysis for asymptotic evaluation.

## Key findings

- Error can be expressed through discrete orthogonal polynomials.
- Asymptotic evaluation of these polynomials is achieved using Riemann-Hilbert methods.
- Provides insights into the accuracy of Fourier extension approximations.

## Abstract

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.