On the resolution power of Fourier extensions for oscillatory functions

Functions that are smooth but non-periodic on a certain interval have only slowly converging Fourier series, due to the Gibbs phenomenon. However, they can be represented accurately by a Fourier series that is periodic on a larger interval. This is commonly called a Fourier extension. Fourier extensions have been mostly used to solve PDE’s on complicated domains, by embedding the domain into a larger bounding box and extending all functions involved to Fourier series on that box, thereby enabling fast FFT-based algorithms. Fourier extensions have also been employed to resolve the Gibbs-phenomenon for non-periodic functions. In this paper we describe, analyze and explain the observation that Fourier extensions also have excellent resolution properties for representing oscillatory functions. The resolution power, or required number of degrees of freedom per wavelength, depends on a usercontrolled parameter and varies between 2 and π. The former value is optimal and is achieved for example by classical Fourier series for periodic functions. The latter value is the resolution power of polynomials. In addition, we also introduce and analyze a new numerical method for computing Fourier extensions, which improves on previous approaches.