TITLE: Bounds on the L^1 norm of a discrete Fourier spectrum
QUESTION [5 upvotes]: I am dealing with a function $f$ of the form
\begin{equation}
f(t):=\sum_{k=1}^Na_ke^{\mathrm{i}\phi_k t}
\end{equation}
and I have a promise that 
\begin{equation}
0\leq f(t)\leq C\;\;\;\text{for all}\;\;\;t\in\mathbb{R},
\end{equation}
where $C>0$ is some constant. My question is the following: What bound can I find for
\begin{equation}
\sum_k|a_k|
\end{equation}
in terms of $C$ and $N$?
Edit: The $\phi_k$ are real numbers and I have the additional promise that $f$ is real. The $a_k$ are complex numbers.

REPLY [5 votes]: To avoid trivialities, I will assume the $\phi_k$ are all distinct. Then
$$ \sum_k |a_k|^2 = \lim_{R \to \infty} \frac{1}{2R} \int_{-R}^R |f(t)|^2\; dt 
\le C^2$$
so by Cauchy-Schwarz, $$\sum_k |a_k| \le C \sqrt{N}$$