Accuracy of the Fourier Extension Method for Oscillatory Phenomena

We introduce and analyse the so-called Fourier extension method for the approximation of oscillatory phenomena in bounded intervals. As we show, this method possesses good resolution properties for such problems. In particular, the resolution constant, the number of degrees of freedom per wavelength required to resolve an oscillation at a given frequency, can be varied between 2 and π by a user-determined parameter. The former value is optimal and achieved by Fourier series, but only when oscillations are periodic. Conversely, the Fourier extension method allows for the resolution of both periodic and nonperiodic oscillations with near-optimal complexity.