"Every function can be represented as a Fourier series"?

Solution 1:

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.

• The function should be piecewise continuous.
• the function should be be bounded.

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.

Solution 2:

I came across this question because I wanted to ask the same thing. In Gilbert Strang's Linear Algebra LEC 24, towards the end: https://youtu.be/8MF3pz-oYHo?t=41m8s he mentions that the parts of a fourier series is like an orthogonal basis, and that you can project a function onto the fourier series like any other projection onto orthogonal basis.

And the intuitive way to think about its restriction to be periodic is because the parts that make up the fourier series are periodic.

(I'd love to see something more rigourous though)

Solution 3:

A bounded periodic integrable function F will certainly "have" a Fourier series, but the sum of the series can fail to be equal to F at some points, even if F is continuous.