Geodesic
A Generalization of Kolmogorov's Theorem to Biorthogonal Systems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and nonlinear partial differential equations, Tome 260 (2008), pp. 44-56.

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

The fundamental Kolmogorov's theorem about divergent trigonometric Fourier series is generalized to bounded biorthonormal systems defined on a separable metric space with Borel regular outer measure. Sharp lower bounds at points and on sets of positive measure are obtained for the arithmetic means of the symmetrized Lebesgue functions of biorthonormal systems defined on an arbitrary measure space. Earlier, similar results were obtained by the author for orthogonal systems on an interval. 
@article{TM_2008_260_a3,
     author = {S. V. Bochkarev},
     title = {A {Generalization} of {Kolmogorov's} {Theorem} to {Biorthogonal} {Systems}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {44--56},
     publisher = {mathdoc},
     volume = {260},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2008_260_a3/}
}
TY  - JOUR
AU  - S. V. Bochkarev
TI  - A Generalization of Kolmogorov's Theorem to Biorthogonal Systems
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2008
SP  - 44
EP  - 56
VL  - 260
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2008_260_a3/
LA  - ru
ID  - TM_2008_260_a3
ER  - 
%0 Journal Article
%A S. V. Bochkarev
%T A Generalization of Kolmogorov's Theorem to Biorthogonal Systems
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2008
%P 44-56
%V 260
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2008_260_a3/
%G ru
%F TM_2008_260_a3
S. V. Bochkarev. A Generalization of Kolmogorov's Theorem to Biorthogonal Systems. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and nonlinear partial differential equations, Tome 260 (2008), pp. 44-56. http://geodesic.mathdoc.fr/item/TM_2008_260_a3/

[1] Kolmogoroff A. N., “Une série de Fourier–Lebesgue divergente presque partout”, Fund. math., 4 (1923), 324–328 | Zbl

[2] Kolmogoroff A. N., “Une série de Fourier–Lebesgue divergente partout”, C. r. Acad. sci. Paris, 183 (1926), 1327–1329

[3] Bochkarev S. V., “Raskhodyaschiisya na mnozhestve polozhitelnoi mery ryad Fure dlya proizvolnoi ogranichennoi ortonormirovannoi sistemy”, Mat. sb., 98:3 (1975), 436–449 | MR | Zbl

[4] Bochkarev S. V., Metod usrednenii v teorii ortogonalnykh ryadov i nekotorye voprosy teorii bazisov, Tr. MIAN, 146, Nauka, M., 1978 | MR | Zbl

[5] Bochkarev S. V., “Metod usrednenii v teorii ortogonalnykh ryadov”, Proc. Intern. Congr. Math., V. 2 (Helsinki, 1978), Acad. Sci. Fenn., Helsinki, 1980, 599–604 | MR

[6] Bochkarev S. V., “Vsyudu raskhodyaschiesya ryady Fure po sisteme Uolsha i multiplikativnym sistemam”, UMN, 59:1 (2004), 103–124 | MR

[7] Bochkarev S. V., “Logarifmicheskii rost srednikh arifmeticheskikh ot funktsii Lebega ogranichennykh ortonormirovannykh sistem”, DAN SSSR, 223:1 (1975), 16–19 | MR | Zbl

[8] Kachmazh S., Shteingauz G., Teoriya ortogonalnykh ryadov, Fizmatgiz, M., 1958 | MR

[9] Bochkarev S. V., “Ob absolyutnoi skhodimosti ryadov Fure po ogranichennym sistemam”, Mat. zametki, 15:3 (1974), 363–370 | Zbl

[10] Kashin B. S., “Zamechaniya ob otsenke funktsii Lebega ortonormirovannykh sistem”, Mat. sb., 106:3 (1978), 380–385 | MR | Zbl

[11] Kashin B. S., Saakyan A. A., Ortogonalnye ryady, Nauka, M., 1984 | MR | Zbl

[12] Bochkarev S. V., “O probleme Zigmunda”, Izv. AN SSSR. Ser. mat., 37:3 (1973), 630–638 | MR | Zbl

[13] Bochkarev S. V., “Ob absolyutnoi skhodimosti ryadov Fure po ogranichennym polnym ortonormirovannym sistemam funktsii”, Mat. sb., 93:2 (1974), 203–217 | Zbl

[14] Bochkarev S. V., “Multiplikativnye neravenstva dlya $L_1$-normy, primeneniya v analize i teorii chisel”, Tr. MIAN, 255, Nauka, M., 2006, 55–70 | MR

[15] Stein E. M., “On limits of sequences of operators”, Ann. Math. Ser. 2, 74 (1961), 140–170 | DOI | MR | Zbl

[16] Saks S., Teoriya integrala, Faktorial, M., 2004

[17] Khalmosh P., Teoriya mery, Faktorial, M., 2003

[18] Federer G., Geometricheskaya teoriya mery, Nauka, M., 1987 | MR | Zbl