Out of the Fourier transform grew an entire field of mathematics, called harmonic analysis, which studies the components of functions.
He spent the next decade conflicted about whether to dedicate his life to religion or to math, eventually abandoning his religious training and becoming a teacher.
Most mathematicians at the time believed that no number of smooth curves could ever add up to a sharp corner.
The process only fails for the most bizarre functions, like those that oscillate wildly no matter how much you zoom in on them.
This infinite set is called the Fourier series, and — despite mathematicians’ early hesitation to accept such a thing — it is now an essential tool in the analysis of functions.
In the 1960s, the mathematicians James Cooley and John Tukey came up with an algorithm that could perform a Fourier transform much more quickly — aptly called the fast Fourier transform.
Mathematicians have also found that harmonic analysis has deep and unexpected connections to number theory.
There are several transformations, such as the Fujisaki-Okamoto (FO) transform, that construct a (secure) KEM from a (secure) PKE scheme.