Fourier series convergence Let $f_n \rightarrow f$ be a sequence of $2\pi$-periodic functions, where the convergence is in $L^1({\mathbb R}/2\pi{\mathbb Z})$.
Then the Fourier-coefficients satisfy $|F(f_n) -F(f)| \rightarrow 0 $ uniformly. 
Now, I was wondering. Does this imply that the Fourier series $$h_k(x):=\sum_{n \in \mathbb{Z}} F(f_k)(n)e^{-inx} $$ converges to $$h(x):=\sum_{n \in \mathbb{Z}} F(f)(n)e^{-inx} $$ pointwise, where we assume that everything exists( so we assume that the sums $h_n,h$ converge for all $x$.)

Let $\{f_n\}$ be the typewriter sequence. $f_n$ converges to $f\equiv0$ in $L^1$ but not pointwise (not even pointwise almost everywhere). The Fourier coefficients of $f$ are all equal to $0$, so its Fourier series converges to $0$. The Fourier series of $f_n$ converges to $f_n$ at all points except two.
