Geodesic
Uniqueness theorems for simple trigonometric series with application to multiple series
Sbornik. Mathematics, Tome 212 (2021) no. 12, pp. 1675-1693 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For simple trigonometric series it is shown, in particular, that if the trigonometric series is Riemann summable in measure to an integrable function $f$ and if the Riemann majorant is finite everywhere except possibly on a countable set, then this series is the Fourier series of the function $f$. Uniqueness theorems for multiple trigonometric series are obtained on the basis of this result. Bibliography: 14 titles.
Keywords: trigonometric system, Riemann summation method, uniqueness theorem. 
@article{SM_2021_212_12_a1,
     author = {G. G. Gevorkyan},
     title = {Uniqueness theorems for simple trigonometric series with application to multiple series},
     journal = {Sbornik. Mathematics},
     pages = {1675--1693},
     year = {2021},
     volume = {212},
     number = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2021_212_12_a1/}
}
TY  - JOUR
AU  - G. G. Gevorkyan
TI  - Uniqueness theorems for simple trigonometric series with application to multiple series
JO  - Sbornik. Mathematics
PY  - 2021
SP  - 1675
EP  - 1693
VL  - 212
IS  - 12
UR  - http://geodesic.mathdoc.fr/item/SM_2021_212_12_a1/
LA  - en
ID  - SM_2021_212_12_a1
ER  - 
%0 Journal Article
%A G. G. Gevorkyan
%T Uniqueness theorems for simple trigonometric series with application to multiple series
%J Sbornik. Mathematics
%D 2021
%P 1675-1693
%V 212
%N 12
%U http://geodesic.mathdoc.fr/item/SM_2021_212_12_a1/
%G en
%F SM_2021_212_12_a1
G. G. Gevorkyan. Uniqueness theorems for simple trigonometric series with application to multiple series. Sbornik. Mathematics, Tome 212 (2021) no. 12, pp. 1675-1693. http://geodesic.mathdoc.fr/item/SM_2021_212_12_a1/

[1] N. K. Bary, A treatise on trigonometric series, v. I, II, A Pergamon Press Book The Macmillan Co., New York, 1964, xxiii+553 pp., xix+508 pp. | MR | MR | Zbl

[2] G. Cantor, “Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen”, Math. Ann., 5:1 (1872), 123–132 | DOI | MR | Zbl

[3] D. Menchoff, “Sur l'unicité du développement trigonometrique”, C. R. Acad. Sci. Paris, 163 (1916), 433–436 | Zbl

[4] A. B. Aleksandrov, “$\mathrm A$-integrability of the boundary values of harmonic functions”, Math. Notes, 30:1 (1981), 515–523 | DOI | MR | Zbl

[5] G. G. Gevorkyan, “On the uniqueness of trigonometric series”, Math. USSR-Sb., 68:2 (1991), 325–338 | DOI | MR | Zbl

[6] G. G. Gevorkyan, “O trigonometricheskikh integralakh, summiruemykh metodom Rimana”, Matem. zametki, 45:5 (1989), 114–117 | MR | Zbl

[7] G. G. Gevorkyan, “On uniqueness of multiple trigonometric series”, Russian Acad. Sci. Sb. Math., 80:2 (1995), 335–365 | DOI | MR | Zbl

[8] Ch. J. Vallee-Poussin, “Sur l'unicité du développement trigonométrique”, Bull. Acad. Roy. Belgique, 1912 (1912), 702–718 | Zbl

[9] G. G. Gevorkyan, “Uniqueness of trigonometric series”, J. Contemp. Math. Anal., 55:6 (2020), 365–375 | DOI | MR | Zbl

[10] J. Marcinkiewicz, A. Zygmund, “On the differentiability of functions and summability of trigonometrical series”, Fundamenta Math., 26 (1936), 1–43 | DOI | Zbl

[11] A. Zygmund, Trigonometric series, v. II, 2nd ed., Cambridge Univ. Press, New York, 1959, vii+354 pp. | MR | MR | Zbl | Zbl

[12] J. M. Ash, G. V. Welland, “Convergence, uniqueness, and summability of multiple trigonometric series”, Trans. Amer. Math. Soc., 163 (1972), 401–436 | DOI | MR | Zbl

[13] Sh. T. Tetunashvili, “On some multiple function series and the solution of the uniqueness problem for Pringsheim convergence of multiple trigonometric series”, Math. USSR-Sb., 73:2 (1992), 517–534 | DOI | MR | Zbl

[14] L. D. Gogoladze, “On the problem of reconstructing the coefficients of convergent multiple function series”, Izv. Math., 72:2 (2008), 283–290 | DOI | DOI | MR | Zbl