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

TL;DR

This paper establishes new L1-norm inequalities for non-harmonic trigonometric polynomials with sparse frequencies, extending classical results to cases with increasing gaps and small observation intervals, with applications to Schrödinger equations.

Contribution

It introduces novel inequalities for sparse frequency sequences with diverging gaps, allowing arbitrarily small observation intervals, extending prior harmonic analysis results.

Findings

Inequalities hold for sequences with diverging gaps.

Results apply to arbitrarily small observation intervals.

Applications demonstrated in Schrödinger equation observability.

Abstract

In this paper we show that, if an increasing sequence $\Lambda=(\lambda_k)_{k\in\mathbb{Z}}$ has gaps going to infinity $\lambda_{k+1}-\lambda_k\to +\infty$ when $k\to\pm\infty$, then for every $T>0$ and every sequence $(a_k)_{k\in\mathbb{Z}}$ and every $N\geq 1$, $$ A\sum_{k=0}^N\frac{|a_k|}{1+k}\leq\frac{1}{T}\int_{-T/2}^{T/2} \left|\sum_{k=0}^N a_k e^{2i\pi\lambda_k t}\right|\,\mbox{d}t$$ further, if $\sum_{k\in\mathbb{Z}}\dfrac{1}{1+|\lambda_k|}<+\infty$,$$ B\max_{|k|\leq N}|a_k|\leq\frac{1}{T}\int_{-T/2}^{T/2} \left|\sum_{k=-N}^N a_k e^{2i\pi\lambda_k t}\right|\,\mbox{d}t $$ where $A,B$ are constants that depend on $T$ and $\Lambda$ 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 $\lambda_{k+1}-\lambda_k\geq 1$. 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 $\lambda_k$'s are integers where it extends a result of McGehee, Pigno and Smith. The results are then applied to observability of Schr\"odinger equations with moving sensors.