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It's interesting to note that Fourier wasn't trying to any of the things that the Fourier Transform is commonly used for today, namely signal processing. He was trying to solve heat transfer equations when he came up with the Fourier series. I'm not even sure if he was that interested in the Transform as such, e.g. looking at a signal in frequency space and then efficiently applying filters before transforming back to time domain.

My dad remembers his professor, sometime in the 40s, posing the question of calculating when a worm buried in the ground would experience the same temperature we'd experience at Christmas (Erdwuermchen's Weihnachten) and the solution had to be calculated with Fourier's heat transfer equations.

when it first came out most people ridiculed it for being a mere intellectual curiosity.

I would characterize the fourier transform (especially the DFT) as the single most important mathematical innovation that enables the interface between the digital and analog world.

I agree although there's a continuum from Fourier to Cooley-Tukey and so on so it's hard to pinpoint which part is the most important.

Also, maybe it's more beloved by engineers (like myself) than mathematicians.

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