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- W2056577268 abstract "We consider distributions $uinmathscr{S}'(mathbb{R})$ of the form $u(t)=sum_{ninmathbb{N}}a_nmathrm{e}^{imath,lambda_{n}t}$, where $(a_n)_{ninmathbb{N}}subsetmathbb{C}$ and $Lambda=(lambda_n)_{ninmathbb{N}}subsetmathbb{R}$ have the following properties: $(a_n)_{ninmathbb{N}}in s'$, that is, there is a $qinmathbb{N}$ such that $(n^{-q},a_n)_{ninmathbb{N}}inell^1$; for the real sequence $Lambda$, there are $n_0inmathbb{N}$, $C>0$, and $alpha>0$ such that $ngeq n_0Rightarrowleft|,lambda_n,right|geq Cn^alpha$. Let $I_epsilonsubsetmathbb{R}$ 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 $u|_{I_epsilon}=0Rightarrow uequiv0$; (2) if $Lambda=mathrm{O}(n)$ is uniformly discrete, then $existsepsilon>0$ such that $u|_{I_epsilon}=0Rightarrow uequiv0$; (3) if $alpha>1$ and $Lambda$ is uniformly discrete, then for all $epsilon>0$, $u|_{I_epsilon}=0Rightarrow uequiv0$. 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." @default.
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- W2056577268 date "2011-01-01" @default.
- W2056577268 modified "2023-09-26" @default.
- W2056577268 title "Uniqueness and Nonuniqueness Results for a Certain Class of Almost Periodic Distributions" @default.
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- W2056577268 doi "https://doi.org/10.1137/090763524" @default.
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