Geodesic
Multiple Walsh Series and Zygmund Sets
Matematičeskie zametki, Tome 95 (2014) no. 5, pp. 750-762

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

The classical Zygmund theorem claims that, for any sequence of positive numbers $\{\varepsilon_n\}$ monotonically tending to zero and any $\delta>0$, there exists a set of uniqueness for the class of trigonometric series whose coefficients are majorized by the sequence $\{\varepsilon_n\}$ whose measure is greater than $2\pi-\delta$. In this paper, we prove the analog of Zygmund's theorem for multiple series in the Walsh system on whose coefficients rather weak constraints are imposed.
Keywords: multiple Walsh series, Zygmund set, set of uniqueness, binary group, Abelian group, binary cube, quasimeasure. 
M. G. Plotnikov. Multiple Walsh Series and Zygmund Sets. Matematičeskie zametki, Tome 95 (2014) no. 5, pp. 750-762. http://geodesic.mathdoc.fr/item/MZM_2014_95_5_a9/
@article{MZM_2014_95_5_a9,
     author = {M. G. Plotnikov},
     title = {Multiple {Walsh} {Series} and {Zygmund} {Sets}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {750--762},
     year = {2014},
     volume = {95},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2014_95_5_a9/}
}
TY  - JOUR
AU  - M. G. Plotnikov
TI  - Multiple Walsh Series and Zygmund Sets
JO  - Matematičeskie zametki
PY  - 2014
SP  - 750
EP  - 762
VL  - 95
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/MZM_2014_95_5_a9/
LA  - ru
ID  - MZM_2014_95_5_a9
ER  - 
%0 Journal Article
%A M. G. Plotnikov
%T Multiple Walsh Series and Zygmund Sets
%J Matematičeskie zametki
%D 2014
%P 750-762
%V 95
%N 5
%U http://geodesic.mathdoc.fr/item/MZM_2014_95_5_a9/
%G ru
%F MZM_2014_95_5_a9

[1] A. Zygmund, “Contribution á l'unicité du développement trigonométrique”, Math. Zeit., 24 (1926), 40–46 | DOI | MR

[2] N. K. Bari, Trigonometricheskie ryady, GIFML, M., 1961 | MR

[3] A. Zigmund, Trigonometricheskie ryady, Mir, M., 1965 | MR

[4] J.-P. Kahane, Y. Katsnelson, “Sur les ensembles d'unicite $U(\varepsilon)$ de Zygmund”, C. R. Acad. Sci., Paris, Sér. A, 277 (1973), 893–895 | Zbl

[5] Sh. T. Tetunashvili, “O edinstvennosti kratnykh trigonometricheskikh ryadov”, Matem. zametki, 58:4 (1995), 596–603 | MR | Zbl

[6] V. L. Shapiro, “$U(\varepsilon)$-sets for Walsh series”, Proc. Amer. Math. Soc., 16:5 (1965), 867–870 | MR | Zbl

[7] A. V. Bakhshetsyan, “Ob $U(\varepsilon)$-mnozhestvakh polnoi mery dlya sistemy Uolsha”, Izv. AN Arm. SSR. Ser. matem., 16 (1981), 431–443 | MR | Zbl

[8] G. G. Gevorkyan, “O mnozhestvakh edinstvennosti dlya sistem Khaara i Uolsha”, Dokl. AN Arm. SSR, 73:2 (1981), 91–96 | Zbl

[9] N. N. Kholschevnikova, “Ob'edinenie mnozhestv edinstvennosti kratnykh ryadov – Uolsha i trigonometricheskikh”, Matem. sb., 193:4 (2002), 135–160 | DOI | MR | Zbl

[10] N. J. Fine, “On the Walsh functions”, Trans. Amer. Math. Soc., 65:3 (1949), 372–414 | DOI | MR | Zbl

[11] G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli, A. I. Rubinshtein, Multiplikativnye sistemy funktsii i garmonicheskii analiz na nulmernykh gruppakh, ELM, Baku, 1981 | MR

[12] B. I. Golubov, A. V. Efimov, V. A. Skvortsov, Ryady i preobrazovaniya Uolsha. Teoriya i primenenie, Nauka, M., 1987 | MR

[13] F. Schipp, W. R. Wade, P. Simon, Walsh Series. An Introduction to Dyadic Harmonic Analysis, Academiai Kiado, Budapest, 1990 | MR

[14] N. Ya. Vilenkin, “Ob odnom klasse polnykh ortonormalnykh sistem”, Izv. AN SSSR. Ser. matem., 11:4 (1947), 363–400 | MR | Zbl

[15] S. Saks, Teoriya integrala, Faktorial Press, M., 2004

[16] M. G. Plotnikov, “O mnozhestvakh edinstvennosti dlya kratnykh ryadov Uolsha”, Matem. zametki, 81:2 (2007), 265–279 | DOI | MR | Zbl