Let’s say we have some periodic function f. The goal is to represent it as a bunch of sine waves. This is called a Fourier series (FS).

The FS coefficients are given by

            ⌠2⋅π
         1  ⎮         -i⋅k⋅x
f̂(k) =  ─── ⎮   f(x)⋅e       dx
        2⋅π ⌡0

for whole numbers k. We can then approximate the original function with

  n
 ___
 ╲          -i⋅k⋅x
 ╱    f̂(k)⋅e
 ‾‾‾
k = -n             

and make the approximation better by making n larger.

For non-periodic functions, we can still get an approximation! The Fourier Transform (FT) uses a continuous range of frequencies instead of whole numbers, and the sum is replaced by an integral (the inverse FT).

FT’s are used everywhere, for example in signal processing and spectroscopy. If a recording contains an annoying high-pitched noise, we can apply the FT and filter the unwanted frequencies. Spectroscopy separates light into its different wavelengths. By comparing the resulting spectral lines with known measurements, we can determine which elements and molecules are present in stars and planets, for example.

idk what more to add ._.