This is almost certainly routine to an analyst, so forgive me in advance.

Let $\alpha_i\in \mathbb{R}$. Consider the functional $$\varphi: L^1[0.9A,A]\to \mathbb{C}$$ via $$f\mapsto \sum_i \hat{f}(\alpha_i),$$ where by '$\hat{f}$' I mean the Fourier transform of $f$ regarded as a function on $\mathbb{R}$ (via extension by zero).

What can one say about the norm of $\varphi$?

I am wondering about the case where all $|\alpha_i| = O(1)$, $A$ is large, and the number of $\alpha_i$ is also large (comparable to $A$). As the title suggests, I'm hoping for a lower bound on $||\varphi||$. The thing that seemed most natural to me was to write this as integrating against a sum of exponentials (say $g(t)$, so that $||g||_\infty = ||\varphi||$) and then calculate the $L^2$ norm of $g$, but I can't bound the off-diagonal terms well enough.

The idea is that if, for some $t$ in the interval, $g(t)$ is very small, then some small-order derivative of $g$ should be bounded below so that $g((1\pm \delta)t)\gg_\delta A^{-O(1)}$ maybe. One could get a lower bound on *some* derivative via the nonvanishing of a Vandermonde determinant, but I think that would require too many derivatives.

Perhaps I am mistaken and there is an example with $||\varphi||\ll \exp(-c A)$?

Anyway, thanks much!