Clarification about equality of Fourier series. Say you have a function, $f(x)$, with period $2\pi$, and say you have computed the Fourier coefficients in the standard way, i.e., 
$$a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) dx,$$
$$a_k = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\cos(nx) dx,$$
$$b_k = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\sin(nx) dx.$$
Then, according to my notes, we say that 
$$f(x) \approx a_0 + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx)).$$
My question is, how would one know if $f(x)$ actually equals this Fourier expansion, instead of just "corresponding" with it? Is there some theorem we can use, instead of actually just summing the expansion, which may be too difficult to do in certain scenarios? 
 A: There are many standard results that apply. Extend your function periodically from $(-\pi,\pi]$ to all of $\mathbb{R}$. For this extended function $f$,


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*(Jordan) If $f$ is of bounded variation on $[a,b]$, then $f$ has left- and right-hand limits at every point of $(a,b)$, and the Fourier series converges pointwise everywhere in $(a,b)$ to the mean of the left- and right-hand limits.

*(Dirichlet) If $f$ has left- and right-hand limits at $x$, and has left- and right-hand derivatives at $x$, then $f$ converges to the mean of the left- and right-hand limits of $f$ at $x$.

*(Fejer) If $f$ has left- and right-hand limits at $x$, then the running average of the Fourier series at $x$ converges to the mean of the left- and right-hand limits of $f$. That is, if
$$
        S_n = a_0 + \sum_{k=1}^{n} a_n\cos(nx)+b_n\sin(nx),
$$
then the sequence of running averages,
$$
     S_0, \frac{S_0+S_1}{2},\frac{S_0+S_1+S_2}{3},\cdots,
$$
converges to the mean of the left- and right-hand limits of $f$. No further smoothness is required.

*(Dini) If there is a number $L$ such that the following holds for some small $\epsilon$, then the Fourier series for $f$ at $x$ converges to $L$:
$$
       \int_{0}^{\epsilon}\frac{1}{u}\left|\frac{f(x-u)+f(x+u)}{2}-L\right|du < \infty
$$

*(Carleson) If $f$ is Lebesgue measurable and square integrable on $[-\pi,\pi]$, then the Fourier series for $f$ converges almost everywhere to $f$.
