Pointwise and uniform convergence of Fourier extensions

by Vincent Coppé and Marcus Webb (KU Leuven)

Location: 200B.02.18, Leuven
Time: Wednesday March 20, 2019 at 10:30

A Fourier extension approximation to a function on [-1,1] is a trigonometric polynomial whose period is T > 1, which removes the periodicity constraint in [-1,1]. In this talk we discuss recent pointwise and uniform convergence results when the approximation is taken to minimise the L^2(-1,1) error. The proof connects Fourier extensions to Legendre polynomials which are orthogonal on an arc of the complex unit circle, and uses Lebesgue’s Lemma for to bound pointwise error — this standard technique in constructive approximation theory involves bounding the Lebesgue function and best uniform approximation estimates. To bound the Lebesgue function we use asymptotics of polynomials orthogonal on an arc of the complex unit circle which were derived by Krasovksy using Riemann-Hilbert analysis. From a computational perspective, Fourier extensions are too ill-conditioned to compute because there is an inherent redundancy. A regularisation is used in practice, leading us to pose some open questions regarding convergence in this practically applicable case.

The talk will be in two 25 minute parts with a short break, and is based on a preprint available at https://arxiv.org/abs/1811.09527.