Geodesic
Uniqueness theorems for Franklin series converging to integrable functions
Sbornik. Mathematics, Tome 209 (2018) no. 6, pp. 802-822 Cet article a éte moissonné depuis la source Math-Net.Ru

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The following results are proved: a) if a Franklin series converges everywhere to a finite integrable function, then it is the Fourier-Franklin series of this function; b) if a Franklin series converges to a finite integrable function everywhere except possibly at points in some countable set and if all its coefficients satisfy a certain necessary condition, then it is the Fourier-Franklin series of this function. Bibliography: 16 titles.
Keywords: Franklin system, uniqueness theorem.
Mots-clés : de la Vallée Poussin theorem 
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G. G. Gevorkyan. Uniqueness theorems for Franklin series converging to integrable functions. Sbornik. Mathematics, Tome 209 (2018) no. 6, pp. 802-822. http://geodesic.mathdoc.fr/item/SM_2018_209_6_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] P. L. Ul'yanov, “Solved and unsolved problems in the theory of trigonometric and orthogonal series”, Russian Math. Surveys, 19:1 (1964), 1–62 | DOI | MR | Zbl

[3] F. G. Arutyunyan, “O edinstvennosti ryadov po sisteme Khaara”, Dokl. AN Arm. SSR, 38:3 (1964), 129–134 | MR | Zbl

[4] M. B. Petrovskaya, “O nul-ryadakh po sisteme Khaara i mnozhestvakh edinstvennosti”, Izv. AN SSSR. Ser. matem., 28:4 (1964), 773–798 | MR | Zbl

[5] V. A. Skvortsov, “Teorema tipa Kantora dlya sistemy Khaara”, Vestn. Mosk. un-ta. Ser. 1 Matem. Mekh., 1964, no. 5, 3–6 | MR | Zbl

[6] G. Faber, “Über die Orthogonalfunktionen des Herrn Haar”, Deutsche Math.-Ver., 19 (1910), 104–112 | Zbl

[7] F. G. Arutyunyan, A. A. Talalyan, “O edinstvennosti ryadov po sistemam Khaara i Uolsha”, Izv. AN SSSR. Ser. matem., 28:6 (1964), 1391–1408 | MR | Zbl

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

[9] Z. Ciesielski, “Properties of the orthonormal Franklin system”, Studia Math., 23:2 (1963), 141–157 | DOI | MR | Zbl

[10] Z. Ciesielski, “Properties of the orthonormal Franklin system. II”, Studia Math., 27:3 (1966), 289–323 | DOI | MR | Zbl

[11] Z. Ciesielski, A. Kamont, “Survey on the orthogonal Franklin system”, Approximation theory, DARBA, Sofia, 2002, 84–132 pp. | MR | Zbl

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

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

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

[15] G. G. Gevorkyan, “O ryadakh po sisteme Franklina”, Anal. Math., 16:2 (1990), 87–114 | DOI | MR | Zbl

[16] G. G. Gevorkyan, “Some theorems on unconditional convergence and a majorant of Franklin series and their application to the spaces $\operatorname{Re}H_p$”, Proc. Steklov Inst. Math., 190 (1992), 49–76 | MR | Zbl