I‘m currently going through my lecture notes and am a bit confused by the different convergence results and would be happy if someone could clarify:)
In the following we always have $f \in L^2((-\pi, \pi), \mathbb{C})$ and $c_k(f):=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(x) e^{-ikx} \ dx$. We define the $N^{th}$ Fourier partial sum as: $S_N(f)(x):=\sum_{k=-N}^N c_k(f)e^{ikx}$.
Now my first result is that with $\mathcal{F}:=\{\frac{e^{ikx}}{\sqrt{s\pi}} | k \in \mathbb{Z}\}$ the span $\operatorname{Span}(\mathcal{F})$, i.e. the Fourier partial sums, are dense in the space of continuous, $2\pi-$periodic functions with respect to the $L^{\infty}$ norm, i.e. the sup norm (up to set of measure zero). Then it follows (since continuous and compactly supported functions are dense in $L^2((-\pi, \pi), \mathbb{C})$), that $\operatorname{Span}(\mathcal{F})$ is also dense in all of $L^2((-\pi, \pi), \mathbb{C})$ w.r.t. the $L^2$ norm. Thus $\mathcal{F}$ is a basis for the Hilbert space $L^2((-\pi, \pi), \mathbb{C})$.
Now my professor remarks that the preceding does not imply that the Fourier sequence of Fourier partial sums converges uniformly to any continuous, compactly supported (on $(-\pi, \pi)$) $f$. I‘m not sure I get that, we have just stated that: $$ \forall \epsilon>0 \ \exists S_N(f) \in \operatorname{Span}(\mathcal{F}): ||f-S_N(f)||_{L^{\infty}}=\operatorname{ess} \sup |f-S_n|=\inf \{M \in \mathbb{R} | \mu(\{x:|f(x)-S_N(f)(x)|>M\})=0\} < \epsilon $$
Haven’t we? This is equivalent to uniform convergence almost everywhere, isn’t it? What’s the problem?
Edit: The only problem I see here is that all coefficients in $S_N(f)$ are allowed to change as $\epsilon$ gets smaller, whereas if we consider uniform convergence „the coefficients with lower indices“ have to stay fixed as $\epsilon$ gets smaller (I hope this makes sense), could that be the issue?
What about pointwise convergence almost everywhere, I know that if a sequence of functions converges in $L^p$ norm then there exists a subsequence that converges pointwise almost everywhere. Wouldn’t that suggest that for all $f \in L^2(-\pi, \pi)$ there is a sequence $(N_j)_{j \in \mathbb{N}}$ s.t.:
$$ f(x)=\lim_{j \rightarrow \infty} \sum_{k=N_j}^{N_j}c_ke^{ikx} $$
for almost every $x \in (-\pi, \pi)$. I fail to see why this doesn’t imply: $$ f(x)=\lim_{N \rightarrow \infty} S_N(f)(x) $$
Shouldn’t these limits be identical? Regarding this my professor remarks that we do indeed have pointwise convergence almost everywhere but that this isn’t a direct consequence from Hilbert space theory (only proved in 1966 by L. Carleson).
So I‘d be happy if someone could clarify my confusion regarding uniform and pointwise convergence:)
