Uniqueness and non-uniqueness results for a certain class of almost periodic distributions
Résumé
We consider distributions $u\in\mathscr{S}'(\mathbb{R})$ of the form $u(t)=\sum_{n\in\mathbb{N}}a_n\mathrm{e}^{\imath\,\lambda_{n}t}$, where $(a_n)_{n\in\mathbb{N}}\subset\mathbb{C}$ and $\Lambda=(\lambda_n)_{n\in\mathbb{N}}\subset\mathbb{R}$ have the following properties: $(a_n)_{n\in\mathbb{N}}\in s'$, that is, there is a $q\in\mathbb{N}$ such that $(n^{-q}\,a_n)_{n\in\mathbb{N}}\in\ell^1$; for the real sequence $\Lambda$, there are $n_0\in\mathbb{N}$, $C>0$, and $\alpha>0$ such that $n\geq n_0\Rightarrow\left|\,\lambda_n\,\right|\geq Cn^\alpha$. Let $I_\epsilon\subset\mathbb{R}$ be an interval of length $\epsilon$. We prove that for given $\Lambda$, (1) if $\Lambda=\mathrm{O}(n^\alpha)$ with $\alpha<1$, then $\nexists\,\epsilon>0$ such that $u|_{I_\epsilon}=0\Rightarrow u\equiv0$; (2) if $\Lambda=\mathrm{O}(n)$ is uniformly discrete, then $\exists\epsilon>0$ such that $u|_{I_\epsilon}=0\Rightarrow u\equiv0$; (3) if $\alpha>1$ and $\Lambda$ is uniformly discrete, then for all $\epsilon>0$, $u|_{I_\epsilon}=0\Rightarrow u\equiv0$. Since distributions of the above mentioned form are very common in engineering, as in the case of the modeling of ocean waves, signal processing, and vibrations of beams, plates, and shells, those uniqueness and nonuniqueness results have important consequences for identification problems in the applied sciences. We show an identification method and close this article with a simple example to show that the recovery of geometrical imperfections in a cylindrical shell is possible from a measurement of its dynamics.