Uniformly convergence of the Fourier Series with coefficients in $\ell^1$

Question

How can I prove these statements?

For each sequence of coefficients $c=\{c_k\} \in\ell^1$, the Fourier series $S_c=\sum_{k \in \mathbb{Z}}c_ke^{ikt}$ converges uniformly to a continuous function. Furthermore, the operator $\ell^1 \longrightarrow C^0([-\pi,\pi])$ defined by $c \mapsto S_c$ is continuous.

I thought about Banach-Steinhaus theorem and the continuity of a limited function between normed spaces but I can't figure out the solution.

Answer 1

Let $e_k\in C[-\pi,\pi]$ be defined by $e_k(t) := e^{ikt}$, then the sum of continuous functions converges absolutely in $\|\cdot\|_\infty$-norm because

$\sum_k \|c_k e_k\|_\infty = \sum_k |c_k|\cdot\|e_k\|_\infty = \sum_k |c_k| = \|c\|_1 < \infty$

This implies uniform convergence $\sum_k c_k e_k = S_c$ for some continuous $S_c\in C[-\pi,\pi]$. The map $c\mapsto S_c$ is continuous because it is linear and the above computation shows

$\|S_c\|_\infty\le\sum_k \|c_k e_k\|_\infty\le\|c\|_1$

Therefore the map has operator norm 1 and thus is continuous.