Fourier transform — why does it even work?

A geometrical interpretation of its inner workings

Image source: Google images and http://firsttimeprogrammer.blogspot.com/2015/04/fourier-series-and-square-wave.html

What this post is about? What you need to know?

A little context

The expansion of Fourier series in terms of sin(nx) and cos(nx)
The expansion of Fourier series in terms of sin(nx) and cos(nx)

Functions as vectors

Vectorially adding up the vector representations of f(x) and g(x) using the fact that f(x)+g(x) for any x is h(x)

Length of function vectors

The dense nature of the vector space

The dot product of function vectors

Vector projections

Orthogonal vectors

The specialty of sines and cosines

The piece-de-resistance

Transition from summation to integrals in the dot product of function vectors

A science enthusiast and passionate philosopher in pursuit of truth.