TITLE: Condition for Fourier series
QUESTION [8 upvotes]: I read that


Any "well-behaved" function of period $2\pi$ can be expressed as a Fourier series. 


What qualifies as "well-behaved"? Any examples of functions that cannot be expressed as a Fourier series?
Thanks.

REPLY [2 votes]: I'll give you one of the easy parts of the answer.  There's the question of what is "well-behaved" and there's the question of what is meant by "expressed as".
Suppose the $n$th partial sum, i.e. the sum of the first $n$ terms of the Fourier series of the function $f$, is $S_n$.  Do you want $\lim\limits_{n\to\infty} S_n = f(x)$?  For every value of $x$?  That's a pretty strong demand, and it can be satisfied for sufficiently well behaved functions $f$.
But suppose you only want $S_n$ to approach $f$ in a weaker sense: you want $\lim\limits_{n\to\infty}\displaystyle\int_0^{2\pi} \left|S_n(x) - f(x)\right|^2\;dx=0.$  That will happen if $f$ is quadratically integrable, i.e. if $\displaystyle\int_0^{2\pi} \left|f(x)\right|^2\;dx<\infty$.
There are lots of more difficult results to prove and some of them had to wait well over a century to get proved.