Parameter selection and numerical approximation properties of Fourier extensions from fixed data

TL;DR

This paper studies how the parameters in Fourier extensions influence approximation accuracy and stability, revealing that fixing the condition number allows flexible parameter choices, notably T=2, with implications for resolution and computational efficiency.

Contribution

It demonstrates that for a fixed condition number, the choice of extension parameters has little impact on accuracy, enabling optimal parameter selection like T=2 for faster algorithms.

Findings

Parameter choice has minimal effect on accuracy at fixed condition number.

T=2 extension is optimal for computational speed.

Resolution power is proportional to T times the oversampling ratio.

Abstract

Fourier extensions have been shown to be an effective means for the approximation of smooth, nonperiodic functions on bounded intervals given their values on an equispaced, or in general, scattered grid. Related to this method are two parameters. These are the extension parameter $T$ (the ratio of the size of the extended domain to the physical domain) and the oversampling ratio $\eta$ (the number of sampling nodes per Fourier mode). The purpose of this paper is to investigate how the choice of these parameters affects the accuracy and stability of the approximation. Our main contribution is to document the following interesting phenomenon: namely, if the desired condition number of the algorithm is fixed in advance, then the particular choice of such parameters makes little difference to the algorithm's accuracy. As a result, one is free to choose $T$ without concern that it is suboptimal. In particular, one may use the value $T=2$ - which corresponds to the case where the extended domain is precisely twice the size of the physical domain - for which there is known to be a fast algorithm for computing the approximation. In addition, we also determine the resolution power (points-per-wavelength) of the approximation to be equal to $T \eta$, and address the trade-off between resolution power and stability.