Dear all,


I would like to know whether the following claim is true. In particular, if it is true, then I would like to know if there is some textbook that contains the statement and maybe even the proof:


**Claim**  

$\forall n \in \mathbb{N}, \ \exists k\_n > 0$ such that


for any real numbers $A\_1, \ldots, A\_n, \varphi\_1, \ldots, \varphi\_n$
and
 any *distinct* positive numbers $\lambda\_1, \ldots, \lambda\_n$,


$\sup\_{t \in \mathbb{R}} \left| \sum\_{i=1}^n A\_i \cos(\lambda\_i t + \varphi\_i) \right| \geq k\_n \max(|A\_1|, \ldots, |A\_n|)$.


I think I have a (quite complicated) proof for $n \leq 3$, but I would like to know whether the statement holds in general.


Many thanks indeed,  

Julian.



Can you control the amplitudes of a finite collection of sine curves just by controlling the amplitude of their superposition?
Yes, this is true. It follows from the fact that the average
$$\lim\_{T\to\infty} {1\over T}\int\_0^T\cos(\lambda\_i t+\phi\_i)\sum\_j A\_j\cos(\lambda\_j t+\phi\_j)\,dt$$
is $A\_i/2$. Clearly the average is bounded by the supremum of the function.


