Completeness of the Trigonometric System for the Classes~$\varphi(L)$
Matematičeskie zametki, Tome 81 (2007) no. 5, pp. 707-712.

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

We obtain a necessary and sufficient condition for the completeness of the trigonometric system with gaps for the classes $\varphi(L)$.
Keywords: trigonometric system with gaps, $2\pi$-periodic function, Jensen's inequality, trigonometric polynomial, Parseval's equality, Riesz–Fischer theorem.
@article{MZM_2007_81_5_a7,
     author = {Yu. S. Kolomoitsev},
     title = {Completeness of the {Trigonometric} {System} for the {Classes~}$\varphi(L)$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {707--712},
     publisher = {mathdoc},
     volume = {81},
     number = {5},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2007_81_5_a7/}
}
TY  - JOUR
AU  - Yu. S. Kolomoitsev
TI  - Completeness of the Trigonometric System for the Classes~$\varphi(L)$
JO  - Matematičeskie zametki
PY  - 2007
SP  - 707
EP  - 712
VL  - 81
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2007_81_5_a7/
LA  - ru
ID  - MZM_2007_81_5_a7
ER  - 
%0 Journal Article
%A Yu. S. Kolomoitsev
%T Completeness of the Trigonometric System for the Classes~$\varphi(L)$
%J Matematičeskie zametki
%D 2007
%P 707-712
%V 81
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2007_81_5_a7/
%G ru
%F MZM_2007_81_5_a7
Yu. S. Kolomoitsev. Completeness of the Trigonometric System for the Classes~$\varphi(L)$. Matematičeskie zametki, Tome 81 (2007) no. 5, pp. 707-712. http://geodesic.mathdoc.fr/item/MZM_2007_81_5_a7/

[1] P. L. Ulyanov, “Predstavlenie funktsii ryadami i klassy $\varphi(L)$”, UMN, 27:2 (1972), 3–52 | MR | Zbl

[2] V. I. Ivanov, “Predstavlenie funktsii ryadami v metricheskikh simmetrichnykh prostranstvakh bez lineinykh funktsionalov”, Vsesoyuznaya shkola po teorii funktsii (Dushanbe, avgust 1986 g.), Tr. MIAN, 189, Nauka, M., 1989, 34–77 | MR | Zbl

[3] V. I. Filippov, “Linear continuous functionals and representation of functions by series in the spaces $E_\varphi$”, Anal. Math., 27:4 (2001), 239–260 | DOI | MR | Zbl

[4] A. A. Talalyan, “Predstavlenie funktsii klassov $L_p[0,1]$, $0

1$, ortogonalnymi ryadami”, Acta Math. Academ. Sci. Hungar., 21:1–2 (1970), 1–9 | DOI | MR | Zbl

[5] K. de Leeuw, “The failure of spectral analysis in $L_p$ for $0

1$”, Bull. Amer. Math. Soc., 82:1 (1976), 111–114 | DOI | MR | Zbl

[6] J. H. Shapiro, “Subspaces of $L_p(G)$ spanned by characters: $0

1$”, Israel J. Math., 29:2–3 (1978), 248–264 | DOI | MR | Zbl

[7] A. B. Aleksandrov, “Essays on non locally convex Hardy classes”, Complex analysis and spectral theory (Leningrad, 1979/1980), Lecture Notes in Math., 864, Springer-Verlag, Berlin, 1981, 1–89 | MR | Zbl

[8] V. I. Ivanov, V. A. Yudin, “O trigonometricheskoi sisteme v $L_p$, $0

1$”, Matem. zametki, 28:6 (1980), 859–868 | MR | Zbl

[9] V. I. Ivanov, “Predstavlenie izmerimykh funktsii kratnymi trigonometricheskimi ryadami”, Ortogonalnye ryady i priblizhenie funktsii, Posv. 100-letiyu so dnya rozhd. akad. N. N. Luzina, Tr. MIAN, 164, Nauka, M., 1983, 100–123 | MR | Zbl

[10] D. Ya. Spivakovskaya, “O trigonometricheskoi sisteme v metricheskikh prostranstvakh $L_\Psi$”, Vestn. Dnepropetr. nats. un-ta. Matem., 2001, no. 6, 101–115

[11] A. Zigmund, Trigonometricheskie ryady, t. 1, 2, Mir, M., 1965 | MR | Zbl

[12] M. A. Krasnoselskii, Ya. B. Rutitskii, Vypuklye funktsii i prostranstva Orlicha, Fizmatgiz, M., 1958 | MR | Zbl

[13] R. M. Trigub, E. S. Belinsky, Fourier Analysis and Approximation of Functions, Kluwer Acad. Publ., Dordrecht, 2004 | MR | Zbl