When an Fourier Analysis should be used for timeseries data

This is a very general question. As a rule, I tend to get more use out of answers when I put more work into asking the question.

Fourier analysis can be used in many ways. I used it, for example, to automatically detect the dominant frequency in experiments.

In cases when you are expecting periodic variation, you can decompose your signal into Fourier series, enforcing period, and then reassemble back the signal - this removes a lot of noise that does not occur with the same period as your signal.

Fourier analysis can be used to take derivatives with respect to time, indeed in Fourier domain this is just multiplication.

These are just few examples.