Geodesic
On Fourier Coefficients of Lacunary Systems
Matematičeskie zametki, Tome 100 (2016) no. 4, pp. 483-491.

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

We prove that the Zygmund space $L(\ln L)^{1/2}$ is the greatest one in the set of symmetric spaces $X$ for which any uniformly bounded orthonormal system of functions contains a sequence such that the corresponding space of Fourier coefficients $F(X)$ coincides with $\ell_2$. Similar results also hold for symmetric spaces located between the spaces $L(\ln L)^{1/2}$ and $L_1$.
Keywords: orthonormal system, symmetric space, real interpolation method.
Mots-clés : Fourier coefficients 
@article{MZM_2016_100_4_a0,
     author = {S. V. Astashkin and E. M. Semenov},
     title = {On {Fourier} {Coefficients} of {Lacunary} {Systems}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {483--491},
     publisher = {mathdoc},
     volume = {100},
     number = {4},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2016_100_4_a0/}
}
TY  - JOUR
AU  - S. V. Astashkin
AU  - E. M. Semenov
TI  - On Fourier Coefficients of Lacunary Systems
JO  - Matematičeskie zametki
PY  - 2016
SP  - 483
EP  - 491
VL  - 100
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2016_100_4_a0/
LA  - ru
ID  - MZM_2016_100_4_a0
ER  - 
%0 Journal Article
%A S. V. Astashkin
%A E. M. Semenov
%T On Fourier Coefficients of Lacunary Systems
%J Matematičeskie zametki
%D 2016
%P 483-491
%V 100
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2016_100_4_a0/
%G ru
%F MZM_2016_100_4_a0
S. V. Astashkin; E. M. Semenov. On Fourier Coefficients of Lacunary Systems. Matematičeskie zametki, Tome 100 (2016) no. 4, pp. 483-491. http://geodesic.mathdoc.fr/item/MZM_2016_100_4_a0/

[1] A. Zigmund, Trigonometricheskie ryady, T. II, Mir, M., 1965 | MR

[2] S. J. Dilworth, “Convergence of series of scalar- and vector-valued random variables and a subsequence principle in $L_2$”, Trans. Amer. Math. Soc., 301:1 (1987), 375–384 | Zbl

[3] S. V. Actashkin, “Sistemy sluchainykh velichin, ekvivalentnye po raspredeleniyu sisteme Rademakhera, i $\mathscr K$-zamknutaya predstavimost banakhovykh par”, Matem. sbornik, 191:6 (2000), 3–30 | DOI

[4] S. G. Krein, Yu. I. Petunin, E. M. Semenov, Interpolyatsiya lineinykh operatorov, Nauka, M., 1978 | MR

[5] V. A. Rodin, E. M. Semyonov, “Rademacher series in symmetric spaces”, Anal. Math., 1:3 (1975), 207–222 | DOI | MR | Zbl

[6] V. A. Rodin, E. M. Semenov, “O dopolnyaemosti podprostranstva, porozhdennogo sistemoi Rademakhera, v simmetrichnom prostranstve”, Funkts. analiz i ego pril., 13:2 (1979), 91–92 | MR | Zbl

[7] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces II. Function Spaces, Springer-Verlag, Berlin, 1979 | MR | Zbl

[8] Yu. A. Brudnyi, N. Ya. Kruglyak, Funktory veschestvennoi interpolyatsii, Dep. v VINITI, No 2620 –81, 1981

[9] Yu. A. Brudnyĭ, N. Ya. Krugljak, Interpolation Functors and Interpolation Spaces, North–Holland, Amsterdam, 1991 | MR | Zbl

[10] I. Berg, I. Lefstrem, Interpolyatsionnye prostranstva. Vvedenie, Mir, M., 1980 | MR

[11] S. Kachmazh, G. Shteingauz, Teoriya ortogonalnykh ryadov, Fizmatgiz, M., 1958

[12] A. Pich, Operatornye idealy, Mir, M., 1982 | MR

[13] B. S. Kashin, A. A. Saakyan, Ortogonalnye ryady, AFTs, M., 1999 | MR

[14] S. V. Astashkin, E. M. Semenov, “Koeffitsienty Fure–Rademakhera”, DAN, 375:1 (2000), 7–9 | MR | Zbl

[15] S. V. Astashkin, “Funktsii Rademakhera v simmetrichnykh prostranstvakh”, Sovrem. matem. Fundam. napr., 32 (2009), 3–161

[16] S. V. Astashkin, “Koeffitsienty Fure–Rademakhera funktsii iz simmetrichnykh prostranstv”, Sib. matem. zhurn., 41:4 (2000), 729–739 | Zbl

[17] N. J. Kalton, “Calderon couples of rearrangement invariant spaces”, Studia Math., 106:3 (1993), 233–277 | MR | Zbl

[18] G. G. Lorentz, T. Shimogaki, “Interpolation theorems for the pairs of spaces $(L^p,L^\infty)$ and $(L^1,L^q)$”, Trans. Amer. Math. Soc., 159 (1971), 207–221 | MR | Zbl

[19] G. Sparr, “Interpolation of weighted $L_p$-spaces”, Studia Math., 62:3 (1978), 229–271 | MR | Zbl

[20] S. V. Astashkin, “O sravnenii funktsii raspredeleniya sluchainykh velichin”, Matem. zametki, 87:1 (2010), 17–25 | DOI | Zbl