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

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