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.



Bounds on the L^1 norm of a discrete Fourier spectrum
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, $$\sum\_k |a\_k| \le C \sqrt{N}$$


