What Is the Fourier Transform?

As everyone in this thread is sharing links, I'm gonna pitch in, too.

This lecture by Dennis Freeman from MIT 6.003 "Signals and Systems" gives an intuitive explanation of the connections between the four popular Fourier transforms (the Fourier transform, the discrete Fourier transform, the Fourier series, and the discrete-time Fourier transform):

https://ocw.mit.edu/courses/6-003-signals-and-systems-fall-2...

Excellent! Thanks!

I wonder what happened to Wavelet transforms? The were very popular years ago, and now one never hears about them.

Really, do you think they've somehow fallen out of favor? If so, that's a surprise to me.

In any case, they are a bit more advanced, and out of scope for the undergraduate course I linked to.

The use-case is slightly different. Wavelets are suited for non-stationary signals, while Fourier Transform has no time localization so it's more for stationary signals. Although short-time Fourier transform exists, which can handle non-stationary signals under the assumption of local stationarity.

Also, a property of wavelets is they're non-parametric, which limits their utility in knowledge discovery applications.

For ML applications, my opinions is that they're somewhat superseded by deep learning methods that apply less restrictive inductive bias. As data grows, the restrictive prior assumptions of wavelets will hurt, sort of like how CNN is being abandoned for ViT, even though CNN can outperform in situations where data is scarce.

So overall, they have a pretty small set of usecases where they're more suited than other alternative tools.

They have specialized applications for sure. I think it's just not as hot an area for new applied math work as 20 years ago.