Can you control the amplitudes of a finite collection of sine curves just by controlling the amplitude of their superposition?

Question

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.

Answer 1

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.