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

Abstract

In this paper we show that, if an increasing sequence $\hat{I}›=(\hat{I}{»}_k{)}_{k\in \mathbb{Z}}$ has gaps going to infinity $\hat{I}{»}_{k+1}-\hat{I}{»}_k\to +\infty$ when $k\to \pm \infty$, then for every $T>0$ and every sequence $(a_k{)}_{k\in \mathbb{Z}}$ and every $N\ge 1$, $$ A\sum_{k=0}^N\frac{|a_k|}{1+k}\leq\frac{1}{T}\int_{-T/2}^{T/2} \left|\sum_{k=0}^N a_k e^{2iπλ_k t}\right|\,\mbox{d}t$$ further, if ${\sum }_{k\in \mathbb{Z}}\frac{1}{1+|\hat{I}{»}_k|}<+\infty$,$$ B\max_{|k|\leq N}|a_k|\leq\frac{1}{T}\int_{-T/2}^{T/2} \left|\sum_{k=-N}^N a_k e^{2iπλ_k t}\right|\,\mbox{d}t $$ where $A,B$ are constants that depend on $T$ and $\hat{I}›$ 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 $\hat{I}{»}_{k+1}-\hat{I}{»}_k\ge 1$. 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 $\hat{I}{»}_k$'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.