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

Abstract

In this paper we show that, if an increasing sequence $Λ=(λ_k)_{k\in\mathbb{Z}}$ has gaps going to infinity $λ_{k+1}-λ_k\to +\infty$ when $k\to\pm\infty$, then for every $T>0$ and every sequence $(a_k)_{k\in\mathbb{Z}}$ and every $N\geq 1$, $$ A\sum_{k=0}^N\frac{|a_k|}{1+k}\leq\frac{1}{T}\int_{-T/2}^{T/2} \left|\sum_{k=0}^N a_k e^{2iπλ_k t}\right|\,\mbox{d}t$$ further, if $\sum_{k\in\mathbb{Z}}\dfrac{1}{1+|λ_k|}<+\infty$,$$ B\max_{|k|\leq N}|a_k|\leq\frac{1}{T}\int_{-T/2}^{T/2} \left|\sum_{k=-N}^N a_k e^{2iπλ_k t}\right|\,\mbox{d}t $$ where $A,B$ are constants that depend on $T$ and $Λ$ only. The first inequality was obtained by Nazarov for $T>1$ and the second one by Ingham for $T\geq 1$ under the condition that $λ_{k+1}-λ_k\geq 1$. 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 $λ_k$'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.