How does the Fourier transform take its input?

Question

I'm using the Fourier command and I'm wondering how does it take its imput. I know that it takes a list of numbers representing a function of time and it gives back a list of numbers representing a function of frequency but I was wondering how is the list represented? For example, if my function goes from $-T$ to $T$ and my list is

list=Table[F[t],{t,-T,T,n}]

will Fourier[list] understand that the first value starts in -T or will assume it's a function going from 0 to $2T$? Also, I have the same question regarding the result. Does the result give a list representing the Fourier transform center at zero? Or does it start with the Fourier transform evaluated at $\omega=0$?

Answer 1

The Fourier function in Mathematica accepts only a rectangular array of data, and not the actual time (or spatial) range. In most of my use cases this means entering a 1D list of data. The resulting transform has no information about the time scale.

The first data point corresponds to a DC offset (0 frequency), the second data point corresponds to a wave that oscillates once across the length of the entire dataset, and so on. My usual method is to generate a list like {0/T, 1/T, 2/T, 3/T, ...} where T is the overall timescale (2T in your case). Then you can get an axis in Hz (or you can do {0*2Pi/T, 1*2Pi/T, ...} if you want angular frequency).

There's a great answer here: What's the correct way to shift zero frequency to the center of a Fourier Transform? if you want to switch to have zero frequency in the middle of the dataset.