/sci/ - Science & Math

The Fourier transform is a unitary operator of L^2 with all kinds of tremendously beautiful properties. It may be defined on any discrete or locally compact abelian group with haar measure, for any appropriately regular measure, for any tempered distribution, on any abelian fucking C*-algebra (the Gelfand transform), etc etc.
It displays truly unimaginable functorial relationships between polynomials and differential operators, between convolution and multiplication, and between a number of other operations. It makes it possible for one to compute all kinds of extremely important quantities in NlogN complexity instead of N^2. It turns partial differential equations, the absolute most difficult kind of mathematical object, into trivially solved algebraic equations. It specializes in the form of Fourier series, allowing essentially any function which you will ever see without trying to make a shitty function with choice to be approximated in many, many topologies by trigonometric polynomials. There is an entire active field of mathematics devoted purely to studying the wide-reaching dualities of the Fourier transform, Harmonic Analysis.
I could go on and on for pages, for days about why the Fourier transform is the most perfect and elegant object ever observed or touched by humans. Nothing else even comes close. It's as if the entirety of logic, mathematics, and the universe was built around the Fourier transform.