Talk:Parallel Spectral Numerical Methods/Finding Derivatives using Fourier Spectral Methods

Untitled
Nullifying the Nyquist term when computing the odd-order derivatives mentioned in the article is very important, and should be emphasized. It is worth also to mention that this is needed for derivatives of functions which are real in the time domain (.i.e., when applying real-to-complex Fourier transform). In the frequency domain, the data for the negative frequencies is complex-conjugate to the data for the corresponding positive frequencies. The Nyquist frequency can be considered both positive and negative (the Nyquist term separates the terms with the positive and negative frequencies), thus the Nyquist term should be complex-conjugate to itself, which means that this term is real (similarly to the dc term). After multiplication by (i omega), the Nyquist term becomes pure imaginary; still, it has to be complex-conjugate to itself, therefore we nullify the Nyquist term when computing the odd-order derivatives.

It is also worth to mention, that if the number of samples is odd, then the Nyquist term does not exist. It seems that in the Matlab numerical example, the number of samples is even (100), but the Nyquist term was not set to zero.

Another important thing is that it may be worth to comment about computing integrals (anti-derivatives) in the Fourier space by dividing by (i omega), except, of course, the dc term with omega = 0. This term can be considered the delta-function, whose inverse transform is the constant value, to be established from the initial conditions in the data space. 76.226.76.104 (discuss) 07:16, 18 January 2021 (UTC)