Suppose $S_N(x)$ is the Fourier series of $f(x)$, a continuous function.
Now, I've understood that if $S_N(x)$ converges uniformly to some $g(x)$ then is must be that $f\equiv g$.
- What about the case where $S_N(x)$ converges pointwise to $g(x)$?
- In which cases $g \not\equiv f$?
Assume $f$ is continuous and $2\pi$-periodic. Then if $S_N$ converges pointwise to some $g$ everywhere, we have $g\equiv f.$ Proof: Recall the result of Fejér: Set
$$F_N(x) = \frac{S_1(x)+ \cdots + S_N(x)}{N}.$$
(So the $F_N$'s are the Cesàro means of the sequence of partial sums of the Fourier series of $f.$) Fejér's theorem says that $F_N \to f$ uniformly on $\mathbb {R}.$
But of course for any fixed $x,$ if $S_N(x)\to L$ then $F_N(x)\to L.$ Since $F_N(x) \to f(x)$ everywhere, we have the result.