A Necessary Condition for the Completeness of the System $\{e^{-\lambda_nt}\mid\operatorname{Re}\lambda_n>0\}$ in the Spaces $C_0(\mathbb R_+)$ and $L^p(\mathbb R_+)$, $p>2$
Abstract:
We obtain a necessary condition for the completeness of the system
$$
e(\Lambda)=\{e^{-\lambda_nt}\mid\operatorname{Re}\lambda_n>0,\,n\in\mathbb Z\}
$$
in
the spaces $C_0$ and $L^p(\mathbb R_+)$, $p>2$, for the case in which the set of limit points of the sequence $\{\lambda_n\}$ is countable and separable.
Keywords:sequence of exponentials, the spaces $C_0(\mathbb R_+)$ and $L^p(\mathbb R_+)$, $p>2$, Szász condition, Hardy class of functions, Bernstein's inequality, analytic function.
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