It seems that some, especially in electrical engineering and musical signal processing, describe that every signal can be represented as a Fourier series.
So this got me thinking about the mathematical proof for such argument.
But even after going through some resources about the Fourier series (which I don't have too much background in, but grasp the concept), I cannot find a mathematical proof for whether every function can be represented by a Fourier series. There was a hint about the function having to be periodic.
So that means that the "every function can be represented as a Fourier series" is a myth and it doesn't apply on signals either, unless they're periodic?
But then I can also find references like these:
http://msp.ucsd.edu/techniques/v0.11/book-html/node171.html
that say/imply that every signal can be made periodic? So does that change the notion about whether Fourier series can represent every function, with the new condition of first making it periodic, if necessary?
Best Answer
Since you're referring to signals here, it seems appropriate to consider this question from the viewpoint of an electrical engineer.
If we impose some restrictions on what kind of functions can be considered a "signal," then all periodic signals have a Fourier series.
These are reasonable physical restrictions that all real signals should meet. These are also more than enough for a function to have a Fourier series.
Now, for a function that isn't periodic, we can find a Fourier series for a piece of it through a process called "windowing." Basically you isolate a part of the signal on some interval, and pretend that piece is one period of a periodic signal. The Fourier coefficients for each "window" tell you the power spectrum of the signal as time progresses.