How/why is it that a periodic function or signal can be split into a series of sinusoids?

Question

The basis of Fourier transformation is the concept that any periodic signal can be broken into frequency and amplitude varying sinusoid's which on adding give the original signal. I find this difficult to digest as I have recently been introduced to Fourier theory. Is there any proof (preferably mathematical) that prove this statement?

Answer 1

For most "reasonable" practical purposes, it is indeed true that the Fourier series "represents" the original function. But/and this requires making "reasonable" and "represents" sufficiently precise. For example, for most continuous functions, the Fourier series fails to converge pointwise to the pointwise values of the original function at many, many points. But we might object that "pointwise values" (at, for example, countably-many points) are not reasonable, in the same way that in real life we cannot measure temperature "at a point", but only on a small region of space. Still, from a neo-platonic mathematical viewpoint, it is arguably of interest (if not "reasonable") to wonder about point-wise values, and much has been written about this.

Pointwise convergence for fairly nice functions was proven c. 1830 by Dirichlet, and perhaps earlier by Fourier in a manuscript that was lost or delayed.

Apart from technicalities about pointwise convergence, we do have $L^2$ convergence, also called "mean-square".

Apart from specific details of convergence, yes, it is fairly remarkable that every periodic function can be represented as a sum of some "basic" periodic functions. This theme was further taken up by Sturm and Liouville in the 1830's, with details fleshed out only in the 1890s by Steklov and Bocher, under the umbrella of "eigenfunction expansions".

Proofs of various of these things are too long to include here, I think, but can be found many places on the internet. (E.g., my notes http://www.math.umn.edu/~garrett/m/fun/)