Geodesic
Systems with nonextendable convergence of quasipolynomials
Matematičeskie zametki, Tome 64 (1998) no. 5, pp. 728-733 
A. A. Ryabinin. Systems with nonextendable convergence of quasipolynomials. Matematičeskie zametki, Tome 64 (1998) no. 5, pp. 728-733. http://geodesic.mathdoc.fr/item/MZM_1998_64_5_a9/
@article{MZM_1998_64_5_a9,
     author = {A. A. Ryabinin},
     title = {Systems with nonextendable convergence of quasipolynomials},
     journal = {Matemati\v{c}eskie zametki},
     pages = {728--733},
     year = {1998},
     volume = {64},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_64_5_a9/}
}
TY  - JOUR
AU  - A. A. Ryabinin
TI  - Systems with nonextendable convergence of quasipolynomials
JO  - Matematičeskie zametki
PY  - 1998
SP  - 728
EP  - 733
VL  - 64
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/MZM_1998_64_5_a9/
LA  - ru
ID  - MZM_1998_64_5_a9
ER  - 
%0 Journal Article
%A A. A. Ryabinin
%T Systems with nonextendable convergence of quasipolynomials
%J Matematičeskie zametki
%D 1998
%P 728-733
%V 64
%N 5
%U http://geodesic.mathdoc.fr/item/MZM_1998_64_5_a9/
%G ru
%F MZM_1998_64_5_a9

Voir la notice de l'article provenant de la source Math-Net.Ru

The system $e(\Lambda)=\bigl\{(it)^ke^{i\lambda_nt}, 0\le k\le m_n-1\bigr\}_{n=1}^\infty$, where $\Lambda=\{\lambda_n\}$ is the set of zeros (of multiplicities $m_n$ ) of the Fourier transform $$ L(z)=\int_{-a}^ae^{izt}\,d\mathscr L(t) $$ of a singular Cantor-Lebesgue measure, is examined. We prove that $e(\Lambda)$ is complete and minimal in $L_p(-a,a)$, with $p\ge1$, and that $|L(x+iy)|^2$ does not satisfy the Muckenhoupt condition on any horizontal line $\operatorname{Im}z=y\ne0$ in the complex plane. This implies that $e(\Lambda)$ does not have the property of convergence extension.

[1] Sedletskii A. M., “Biortogonalnye razlozheniya funktsii v ryady eksponent na intervalakh veschestvennoi osi”, UMN, 37:5 (1982), 51–95 | MR

[2] Kahane J.-P., Salem R., Ensembles parfaits et séries trigonomètriques, Hermann, Paris, 1963 | Zbl

[3] Zigmund A., Trigonometricheskie ryady, T. 1, Mir, M., 1965

[4] Garnett Dzh., Ogranichennye analiticheskie funktsii, Mir, M., 1984 | Zbl

[5] Pavlov B. S., “Bazisnost sistemy eksponent i uslovie Makenkhaupta”, Dokl. AN SSSR, 247:1 (1979), 37–40 | MR | Zbl

[6] Gubreev G. M., “Spektralnyi analiz operatora differentsirovaniya i uslovie Makenkhaupta”, Dokl. AN SSSR, 278:5 (1984), 1052–1056 | MR | Zbl

[7] Levin B. Ya., Raspredelenie kornei tselykh funktsii, GITTL, M., 1956

[8] Ryabinin A. A., “Asimptotika i nuli preobrazovaniya Fure–Stiltesa finitnykh mer Kantora–Lebega”, Dokl. RAN, 342:6 (1995), 742–743 | MR | Zbl