Abstract:
We obtain a result concerning the basis property
in a weighted space on an interval $(-a,a)$
for a system of exponentials generated by the zeros
of the Fourier transform of a function with singularities
at the ends of the support interval $(-a,a)$.
For an arbitrary $\Delta\in\mathbb{C}$ we find a criterion
for the basis property of the system
$(e^{i(n+\Delta\operatorname{sign} n)t})_{n\in\mathbb{Z}}$
in a weighted space on the interval $(-\pi,\pi)$ and the systems
of sines $(\sin((n+\Delta)t))_{n\in\mathbb{N}}$ and cosines
$1\cup (\cos((n+\Delta)t))_{n\in\mathbb{N}}$ in a weighted
space on the interval $(0,\pi)$. The weight is everywhere
a finite product of polynomial functions.
\Bibitem{Puk11}
\by S.~S.~Pukhov
\paper Bases of exponentials, sines and cosines in weighted spaces on a~finite interval
\jour Izv. Math.
\yr 2011
\vol 75
\issue 2
\pages 413--443
\mathnet{http://mi.mathnet.ru/eng/im4203}
\crossref{https://doi.org/10.1070/IM2011v075n02ABEH002539}
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Linking options:
https://www.mathnet.ru/eng/im4203
https://doi.org/10.1070/IM2011v075n02ABEH002539
https://www.mathnet.ru/eng/im/v75/i2/p195
This publication is cited in the following 7 articles:
M. I. Ismailov, I. F. Aliyarova, “On the basis property of the system of exponentials and trigonometric systems of sine and cosine functions in weighted grand Lebesgue spaces”, Moscow University Mathematics Bulletin, 79:2 (2024), 85–97
Yu. A. Kriksin, V. F. Tishkin, “On an approximation by band-limited functions”, Dokl. Math., 110:3 (2024), 500–505
Gasymov T.B., Akhtyamov A.M., Ahmedzade N.R., “On the Basicity of Eigenfunctions of a Second-Order Differential Operator With a Discontinuity Point in Weighted Lebesgue Spaces”, Proc. Inst. Math. Mech., 46:1 (2020), 32–44
V. R. Fatalov, “Integrals of Bessel processes and multi-dimensional Ornstein–Uhlenbeck processes:
exact asymptotics for $L^p$-functionals”, Izv. Math., 82:2 (2018), 377–406
Shukurov A.Sh., “On Basicity of the Degenerate Trigonometric System With Excess”, Acta Comment. Univ. Tartu. Math., 21:2 (2017), 249–257
S. S. Pukhov, “Bases of Exponentials in Weighted Spaces Generated by Zeros of the Fourier–Stieltjes Transform”, Math. Notes, 95:6 (2014), 820–832
V. R. Fatalov, “Gaussian Ornstein–Uhlenbeck and Bogoliubov processes: asymptotics of small deviations for $L^p$-functionals, $0<p<\infty$”, Problems Inform. Transmission, 50:4 (2014), 371–389
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