Geodesic
Completeness and Basis Properties of Systems of Exponentials in Weighted Spaces $L^p(-\pi,\pi)$
Matematičeskie zametki, Tome 81 (2007) no. 5, pp. 776-788.

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We consider the system of exponentials $e(\Lambda)=\{e^{i\lambda_nt}\}_{n\in\mathbb Z}$, where $$ \lambda_n=n+\biggl(\frac{1+\alpha}p+l(|n|)\biggr)\operatorname{sign}n, $$ $l(t)$ is a slowly varying function, and $l(t)\to 0$, $t\to\infty$. We obtain an estimate for the generating function of the sequence $\{\lambda_n\}$ and, with its help, find a completeness criterion and a basis condition for the system $e(\Lambda)$ in the weight spaces $L^p(-\pi,\pi)$. We also study some special cases of the function $l(t)$.
Keywords: system of exponentials, completeness of a system of functions, the weight spaces $L^p(-\pi,\pi)$, Cauchy's theorem, Riesz basis, generating function.
Mots-clés : Laplace transform 
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A. A. Yukhimenko. Completeness and Basis Properties of Systems of Exponentials in Weighted Spaces $L^p(-\pi,\pi)$. Matematičeskie zametki, Tome 81 (2007) no. 5, pp. 776-788. http://geodesic.mathdoc.fr/item/MZM_2007_81_5_a15/

[1] N. Levinson, Gap and Density Theorems, Amer. Math. Soc. Colloq. Publ., 26, Amer. Math. Soc., New York, 1940 | MR | Zbl

[2] A. M. Sedletskii, “Polnye i ravnomerno minimalnye sistemy pokazatelnykh funktsii v $L^p(-\pi,\pi)$”, Tr. MEI, 1972, no. 146, 167–174

[3] R. Redheffer, R. M. Young, “Completeness and basis properties of complex exponentials”, Trans. Amer. Math. Soc., 277:1 (1983), 93–111 | DOI | MR | Zbl

[4] A. I. Kheifits, “Kharakteristiki nulei nekotorykh spetsialnykh klassov tselykh funktsii konechnoi stepeni”, Teor. funktsii, funktsion. anal. i ikh prilozh., t. 9, Kharkov, 1968, 3–13

[5] A. M. Sedletskii, “Nonharmonic analysis”, Functional analysis, J. Math. Sci. (N. Y.), 116:5 (2003), 3551–3619 | DOI | MR | Zbl

[6] A. M. Sedletskii, Klassy analiticheskikh preobrazovanii Fure i eksponentsialnye approksimatsii, Fizmatlit, M., 2005 | MR

[7] E. Seneta, Pravilno menyayuschiesya funktsii, Nauka, M., 1985 | MR | Zbl

[8] G. M. Fikhtengolts, Kurs differentsialnogo i integralnogo ischisleniya, t. 2, Nauka, M., 1966 | MR | Zbl