[bisby]

In my engineering courses, Fourier transforms were taught in the context of discrete fourier transforms. Because sampling is a thing that matters (computer audio is discrete data points, not an actual wave).

The Fourier transform of a discrete signal repeats in the frequency domain. For example, [1, -1, 1] could be a sine wave with the exact half of the sampling frequency going from 1 to -1 back to 1 once .... Or it could be a sine wave with double the sampling frequency that is actually going from 1 to -1 to 1 to -1 all within the gap of the first 2 samples. Or it could be 3x the sampling frequency, 4x the sampling frequency, etc. The solution is to only keep the part of the transform that is below the Nyquist limit, because we don't have a sampling rate accurate enough to measure the higher frequencies, so just assuming they dont exist. This also means that if the source signal WAS in fact 4x the sampling frequency, we will see a spike at 1/2 the sampling frequency in the fourier transform, and when we re-create the signal, it will be completely wrong.

So unless you have analog hardware for measuring the Fourier transform (or are working purely in a non-physical mathematical domain, like "i have a sine wave" which can be perfectly represented), you are naturally going to be taking discrete samples of a signal to measure the Fourier transform, which means you are going to be losing any part of the signal that doesn't adhere to sampling rates.

Because my engineering courses were so heavily focused on digital signal processing, when I hear "fourier transform" i immediately think of "discrete fourier transform" and loss is immediately applicable.

[mitthrowaway2]

Don't engineers at least learn to solve the heat equation analytically using Fourier transforms? It wasn't all numerical algorithms, was it?