Completeness of systems of complex exponentials and the Lambert 𝑊 functions

Boivin, André; Zhong, Hualiang

doi:10.1090/S0002-9947-06-03950-X

Completeness of systems of complex exponentials and the Lambert $W$ functions
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by André Boivin and Hualiang Zhong PDF

Trans. Amer. Math. Soc. 359 (2007), 1829-1849 Request permission

Abstract:

We study some of the properties of the solution system $\{e^{i\lambda _nt}\}$ of the delay-differential equation $y’(t) = ay(t-1)$. We first establish some general results on the stability of the completeness of exponential systems in $L^2$ and then show that the solution system above is always complete, but is not an unconditional basis in $L^2(-1/2,1/2)$.

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