Periodic PS methods are almost always implemented with use of the FFT algorithm. For nonperiodic PS methods, direct matrix × vector multiplication is often both fast and convenient. However, in the case of Chebyshev-PS methods, a cosine-FFT approach is also effective. Following a description of the FFT concept in Section F.1, its use for periodic and Chebyshev- PS implementations is described in Sections F.2 and F.3.

In most periodic PS contexts, what is actually needed is not Fourier expansion coefficients but rather a fast way to compute periodic convolutions. FFTs offer one way to do this. In Section F.4, we discuss convolutions and some alternative ways to calculate them effectively. In Section F.5, we find that, at four times the cost of a “basic” FFT, the scope of the algorithm can be greatly broadened. These fractional Fourier transforms apply to many problems of physical and computational interest.