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. 
@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},
     publisher = {mathdoc},
     volume = {95},
     number = {5},
     year = {2014},
     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
PB  - mathdoc
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
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2014_95_5_a9/
%G ru
%F MZM_2014_95_5_a9
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/

[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