Geodesic
On uniqueness for series in the general Franklin system
Sbornik. Mathematics, Tome 215 (2024) no. 3, pp. 308-322 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove some uniqueness theorems for series in general Franklin systems. In particular, for series in the classical Franklin system our result asserts that if the partial sums $S_{n_i}(x)=\sum_{k=0}^{n_i}a_kf_k(x)$ of a Franklin series $\sum_{k=0}^{\infty}a_kf_k(x)$ converge in measure to an integrable function $f$ and $\sup_i|S_{n_i}(x)|<\infty$, for $x\notin B$, where $B$ is some countable set and $\sup_i(n_i/n_{i-1})<\infty$, then this is the Fourier–Franklin series of $f$. Bibliography: 29 titles.
Keywords: Franklin system, Franklin series, general Franklin system, uniqueness theorem, Fourier–Franklin series. 
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     title = {On uniqueness for series in the general {Franklin} system},
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G. G. Gevorkyan. On uniqueness for series in the general Franklin system. Sbornik. Mathematics, Tome 215 (2024) no. 3, pp. 308-322. http://geodesic.mathdoc.fr/item/SM_2024_215_3_a1/

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

[2] 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 | Zbl

[3] F. G. Arutyunyan, “Series in the Haar system”, Dokl. Akad. Nauk Arm. SSR, 42:3 (1966), 134–140 (Russian) | MR | Zbl

[4] M. B. Petrovskaya, “Null series in the Haar system and uniqueness sets”, Izv. Akad. Nauk SSSR Ser. Mat., 28:4 (1964), 773–798 (Russian) | MR | Zbl

[5] V. A. Skvortsov, “A Cantor-type theorem for the Haar system”, Vestn. Moskov. Univ. Ser. 1 Mat. Mekh., 1964, no. 5, 3–6 (Russian) | MR | Zbl

[6] F. G. Arutyunyan and A. A. Talalyan, “Uniqueness of series in the Haar and Walsh systems”, Izv. Akad. Nauk SSSR Ser. Mat., 28:6 (1964), 1391–1408 (Russian) | MR | Zbl

[7] G. Kozma and A. Olevskiĭ, “Cantor uniqueness and multiplicity along subsequences”, St. Petersburg Math. J., 32:2 (2021), 261–277 | DOI | MR | Zbl

[8] N. N. Kholshchevnikova, “Sum of everywhere convergent trigonometric series”, Math. Notes, 75:3 (2004), 439–443 | DOI | MR | Zbl

[9] V. A. Skvortsov and N. N. Kholshchevnikova, “Comparison of two generalized trigonometric integrals”, Math. Notes, 79:2 (2006), 254–262 | DOI | MR | Zbl

[10] G. G. Gevorkyan, “Uniqueness theorems for simple trigonometric series with application to multiple series”, Sb. Math., 212:12 (2021), 1675–1693 | DOI | MR | Zbl

[11] M. G. Plotnikov, “$\lambda$-Convergence of multiple Walsh–Paley series and sets of uniqueness”, Math. Notes, 102:2 (2017), 268–276 | DOI | MR | Zbl

[12] M. G. Plotnikov and Yu. A. Plotnikova, “Decomposition of dyadic measures and unions of closed $\mathscr{U}$-sets for series in a Haar system”, Sb. Math., 207:3 (2016), 444–457 | DOI | MR | Zbl

[13] G. G. Gevorkyan and K. A. Navasardyan, “Uniqueness theorems for generalized Haar systems”, Math. Notes, 104:1 (2018), 10–21 | DOI | MR | Zbl

[14] G. G. Gevorkyan and K. A. Navasardyan, “Uniqueness theorems for series by Vilenkin system”, J. Contemp. Math. Anal., 53:2 (2018), 88–99 | DOI | MR | Zbl

[15] G. G. Gevorkyan, “Uniqueness theorems for series in the Franklin system”, Math. Notes, 98:5 (2015), 847–851 | DOI | MR | Zbl

[16] G. G. Gevorkyan, “On the uniqueness of series in the Franklin system”, Sb. Math., 207:12 (2016), 1650–1673 | DOI | MR | Zbl

[17] G. G. Gevorkyan, “Uniqueness theorems for Franklin series converging to integrable functions”, Sb. Math., 209:6 (2018), 802–822 | DOI | MR | Zbl

[18] G. G. Gevorkyan, “Ciesielski and Franklin systems”, Approximation and probability, Banach Center Publ., 72, Polish Acad. Sci. Inst. Math., Warsaw, 2006, 85–92 | DOI | MR | Zbl

[19] Z. Wronicz, “On a problem of Gevorkyan for the Franklin system”, Opuscula Math., 36:5 (2016), 681–687 | DOI | MR | Zbl

[20] Z. Wronicz, “Uniqueness of series in the Franklin system and the Gevorkyan problems”, Opuscula Math., 41:2 (2021), 269–276 | DOI | MR | Zbl

[21] G. G. Gevorkyan, “On uniqueness for Franklin series with a convergent subsequence of partial sums”, Sb. Math., 214:2 (2023), 197–209 | DOI | MR | Zbl

[22] G. G. Gevorkyan and V. G. Mikaelyan, “Uniqueness of series by general Franklin system with convergent subsequence of partial sums”, J. Contemp. Math. Anal., 58:2 (2023), 67–80 | DOI | Zbl

[23] G. G. Gevorkyan and A. Kamont, On general Franklin systems, Dissertationes Math. (Rozprawy Mat.), 374, Polish Acad. Sci. Inst. Math. Inst. Math., Warsaw, 1998, 59 pp. | MR | Zbl

[24] Ph. Franklin, “A set of continuous orthogonal functions”, Math. Ann., 100:1 (1928), 522–529 | DOI | MR | Zbl

[25] Z. Ciesielski and A. Kamont, “Projections onto piecewise linear functions”, Funct. Approx. Comment. Math., 25 (1997), 129–143 | MR | Zbl

[26] G. G. Gevorkyan and A. Kamont, “Unconditionality of general Franklin systems in $L^p[0,1]$, $1

\infty$”, Studia Math., 164:2 (2004), 161–204 | DOI | MR | Zbl

[27] V. G. Mikaelyan, “On a ‘martingale property’ of series with respect to general Franklin system”, Dokl. Nats. Akad. Nauk Armen., 120:2 (2020), 110–114 (Russian) | MR

[28] G. G. Gevorkyan, “On a ‘martingale property’ of Franklin series”, Anal. Math., 45:4 (2019), 803–815 | DOI | MR | Zbl

[29] M. de Guzmán, Differentiation of integrals in $\mathbb{R}^n$, Lecture Notes in Math., 481, Springer-Verlag, Berlin–New York, 1975, xii+266 pp. | DOI | MR | Zbl