[antognini]

I am reminded somewhat of a line in Sanjeev Arora's lecture notes A Theorist's Toolkit:

"Sanjeev admits that he used to find Fourier transforms intimidating as a student. His fear vanished once he realized that it is a rewording of the following trivial idea:

If u_1, u_2, ..., u_n is an orthonormal basis of R^n then every vector v can be expressed as Sum_i alpha_i u_i where alpha_i = <v, u_i> and Sum_i alpha_i^2 = |v|^2"

https://www.cs.princeton.edu/~arora/pubs/toolkit.pdf

[ajkjk]

That's not a good enough statement to really summarize Fourier transforms, I think? It really just summarizes the idea of an orthogonal basis.