$h_k(x):=\sum_{n \in \mathbb{Z}} F(f_k)(n)e^{-inx} \rightarrow h(x):=\sum_{n \in \mathbb{Z}} F(f)(n)e^{-inx} $?

Question

Let $f_n \rightarrow f$ be a sequence of functions in the $L^1$ sense. Then the Fourier transform implies $||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 $h_n,h$ are finite for all $x$.)