I generally agree with the point of the article ("Fourier transform is not magical").

However saying it is "just" curve fitting with sinusoids fails to mention that, among an infinite number of basis functions, there are some with useful properties, and sinusoids are one such: they are eigenvectors of shift-invariant linear systems (and hence are also eigenvectors of derivative operators).

All quite good examples but I would say that these are quite well known. It’s also missing that there are mitigation strategies for some - for e.g. in vibration analysis it’s typical to look at the Hann windowed data to remove the effect of partial cycles, and it’s common to overlap samples too. Similarly there are other tools like the Cepstrum which help you identify periodic peaks in the spectral data.

I made a video about a cool application of the Discrete Fourier Transform regarding color eink Kaleido 3 and manga:

https://youtu.be/Dw2HTJCGMhw?si=Qhgtz5i75v8LwTyi

Learning about Fourier is really interesting in image processing, I'm glad I found a good textbook explaining it.

It's not just curve fitting because basis functions have characteristics which make them desirable for the kind of decomposition one is trying to find. We typically assume in factor analysis that factors are gaussian random variables without clear and repeating patterns. Fourrier transforms force us to think in similar terms but accounting for specific dynamics factor (I. E. Basis functions) should capture.

Also how do we construct those orthogonal basis functions for any downstream task is an interesting research question!

"Few people appreciate statistics. But at least they seem OK with this and don't go off starting religious wars over the subject."

Frequentist vs Bayesian get debated constantly. I liked this video about the difference:

https://youtu.be/9TDjifpGj-k?si=BpjlTCWIFMu506VL