The term “Fourier extensions” refers to a technique for approximating nonsmooth or nonperiodic functions using Fourier modes. Fourier series can approximate a smooth and periodic function very well with spectral convergence. With the Cooley-Tukey fast Fourier transform (FFT) algorithm, the cost of such approximation is reduced to O(N log N) from O (N²). However, the picture is not the same for nonsmooth periodic functions.

If one tries to approximate such functions using Fourier series, there would be jumps near nonsmooth parts. This is the well-known Gibbs phenomenon, which creates numerical problems for such functions. Refining the discretization would not solve the problem either, as one cannot get L^∞ convergence. For the case of nonperiodic functions, one cannot use the Fourier series altogether. However, the Fourier extension is an interesting method that resolves these issues by approximating the function on an extended domain. It can be proved that the approximation would converge to the function in the original domain. Now the questions are: How large should the domain be extended? What should be the oversampling ratio? These questions are studied numerically in this paper.

There is a fast way to compute the approximation if the domain is extended to twice its range. That is usually the favored choice, because of its optimal computational cost. But it is important to study how this choice affects the accuracy and stability of the approximation. The conclusion of this paper is interesting. It argues that although T=2 (the extension parameter) may not be the optimal choice for approximating a given function, it makes very little difference given a target condition number. The authors would then study how the choice of oversampling ratio affects the accuracy and stability of the method.

The audience for this paper is people who are already familiar with the Fourier extensions method and its stability and accuracy issues. Although I would have liked to see more in-depth mathematical explanations, the numerical experiments performed are informative. I would recommend this paper to people working in the Fourier extensions field.