On L1-norms for non-harmonic trigonometric polynomials with sparse frequencies

In this paper we show that, if an increasing sequence $\mathrm{\Lambda }=({\lambda }_k{)}_{k\in \mathbb{Z}}$ has gaps going to infinity ${\lambda }_{k+1}-{\lambda }_k\to +\mathrm{\infty }$ when $k\to \pm \mathrm{\infty }$, then for every $T>0$ and every sequence $(a_k{)}_{k\in \mathbb{Z}}$ and every $N\ge 1$, $$A\sum _{k=0}^N\frac{\mathrm{\mid }{a}_k\mathrm{\mid }}{1+k}\le \frac{1}{T}{\int }_{-T\mathrm{/}2}^{T\mathrm{/}2}\mid \sum _{k=0}^N{a}_k{e}^{2i\pi {\lambda }_{k}t}\mid dt$$ further, if ${\sum }_{k\in \mathbb{Z}}{\displaystyle \frac{1}{1+\mathrm{\mid }{\lambda }_k\mathrm{\mid }}}<+\mathrm{\infty }$,$$B\underset{\mathrm{\mid }k\mathrm{\mid }\le N}{\max }\mathrm{\mid }{a}_k\mathrm{\mid }\le \frac{1}{T}{\int }_{-T\mathrm{/}2}^{T\mathrm{/}2}\mid \sum _{k=-N}^N{a}_k{e}^{2i\pi {\lambda }_{k}t}\mid dt$$ where $A,B$ are constants that depend on $T$ and $\mathrm{\Lambda }$ only. The first inequality was obtained by Nazarov for $T>1$ and the second one by Ingham for $T\ge 1$ under the condition that ${\lambda }_{k+1}-{\lambda }_k\ge 1$. The main novelty is that if those gaps go to infinity, then $T$ can be taken arbitrarily small. The result is new even when the ${\lambda }_k$’s are integers where it extends a result of McGehee, Pigno and Smith. The results are then applied to observability of Schrödinger equations with moving sensors.