one-side estimates for quasi-trigonometric polynomial

Question

Let $f(x)=\Re(\sum_{k=1}^n a_k e^{i\lambda_k x})$ for $0 < \lambda_1 < \lambda_2 < \dots < \lambda_n$ and some complex $a_1$, $a_2$, $\dots$, $a_n$. What is the best (in some sense) estimate for $\inf_{[-M,M]} f(x)$ for large $M$ (in particular, for $M=+\infty$). For example, is it true that $\inf f(x)\leq -C|a_1|$ for some absolute $C$?

The best estimate I was able to get contains $\sum |a_i|$ in denominator, but I would like to have the one which does not tend to zero when we add many new terms.

Answer 1

Let's do the case of full line, which is nice and clean. Without loss of generality, $\lambda_1=1$. Then you can ignore all non-integer $\lambda$s because if $f=g+h$ where $g$ includes all integer frequences and $h$ all non-integer ones, then $\inf \Re g$ is the same as the infimum of $\lim_{T\to+\infty}\Re \frac 1{2T}\int_{-T}^T gP$ where $P$ runs over positive $2\pi$-periodic trigonometric polynomials with integral $1$ over the period. Note that the corresponding limit for $h$ is $0$, and that this integral infimum (with $f$ instead of $g$, of course) estimates the infimum of $\Re f$ from above for any $f$ so $\inf\Re f\le\inf\Re g$. Now, once we are in the periodic setting, we can go to the circle and see that our problem is equivalent to the following: given an analytic function $F$ in the unit disk with $F(0)=0$ and $\Re F(z)\ge -1$, estimate $|F'(0)|$ from above. The answer now is given by the conformal mapping (Schwarz lemma).

You can do similar things for finite intervals, but you need some separation assumptions, etc. to get something meaningful. I'll not go into that now just because I'm not sure what exactly you want there and what you can afford to assume.

Answer 2

Just a hint for a result somehow in the direction you are looking at, assuming generic conditions. For a generic choice of $\lambda:=(\lambda_1\,\dots,\lambda_n),$ the vector $\big(\frac{2\pi}{\lambda_1}\,\dots,\frac{2\pi}{\lambda_n}\big)$ generates a dense additive subgroup in $\mathbb{R}^n/\mathbb{Z}^n$. In such a situation we have an equality: $$\sup_\mathbb{R} f=-\inf_\mathbb{R} f=\|f\|_{\infty,\mathbb{R}}= |a_1|+\dots+|a_n|,$$

and in particular the answer to the second question is affirmative with $C=1$.

Rmk: here the assumption on $\lambda$ is "generic" both in topological and measurable sense: precisely, it holds for a $G_\delta$ set of full Lebesgue measure: the complement is a countable union of codimension 1 submanifolds.