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JOURNALS // Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya

Izv. RAN. Ser. Mat., 2022, Volume 86, Issue 6, Pages 161–186 (Mi im9263)

Uniqueness sets of positive measure for the trigonometric system
M. G. Plotnikov

References

1. G. Kantor, Trudy po teorii mnozhestv, per. s nem., Klassiki nauki, Nauka, M., 1985, 431 pp.  mathscinet  zmath
2. A. S. Kechris, A. Louveau, Descriptive set theory and the structure of sets of uniqueness, London Math. Soc. Lecture Note Ser., 128, Cambridge Univ. Press, Cambridge, 1987, viii+367 pp.  crossref  mathscinet  zmath
3. R. Cooke, “Uniqueness of trigonometric series and descriptive set theory, 1870–1985”, Arch. Hist. Exact Sci., 45:4 (1993), 281–334  crossref  mathscinet  zmath
4. A. Zygmund, Trigonometric series, v. I, Cambridge Math. Lib., 3rd ed., Cambridge Univ. Press, Cambridge, 2002, xiv+383 pp.  mathscinet  mathscinet  zmath  zmath
5. 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.  mathscinet  mathscinet  zmath
6. N. N. Kholshchevnikova, “On the de la Vallé-Poussin theorem on the uniqueness of the trigonometric series representing a function”, Sb. Math., 187:5 (1996), 767–784  mathnet  crossref  crossref  mathscinet  zmath
7. J.-P. Kahane, Y. Katznelson, “Sur les ensembles d'unicité $U(\varepsilon)$ de Zygmund”, C. R. Acad. Sci. Paris Sér. A-B, 277 (1973), A893–A895  mathscinet  zmath
8. M. G. Plotnikov, “Recovery of integrable functions and trigonometric series”, Sb. Math., 212:6 (2021), 843–858  mathnet  crossref  crossref  mathscinet  zmath  adsnasa
9. S. B. Stechkin, P. L. Ulyanov, “O mnozhestvakh edinstvennosti”, Izv. AN SSSR. Ser. matem., 26:2 (1962), 211–222  mathnet  mathscinet  zmath
10. J. E. Coury, “Some results on lacunary Walsh series”, Pacific J. Math., 45:2 (1973), 419–425  crossref  mathscinet  zmath
11. S. F. Lukomskiĭ, “Necessary conditions for sets of uniqueness of Walsh series with gaps”, Sb. Math., 61:2 (1988), 461–470  mathnet  crossref  mathscinet  zmath
12. S. V. Astashkin, R. S. Sukhanov, “On certain properties of Rademacher chaos”, Math. Notes, 91:5 (2012), 613–624  mathnet  crossref  crossref  mathscinet  zmath
13. M. Plotnikov, “On the Vilenkin–Chrestenson systems and their rearrangements”, J. Math. Anal. Appl., 492:1 (2020), 124391, 13 pp.  crossref  mathscinet  zmath
14. G. Kozma, A. M. Olevskiĭ, “Cantor uniqueness and multiplicity along subsequences”, Algebra i analiz, 32:2 (2021), 85–106  mathnet  mathscinet  zmath
15. P. L. Ul'yanov, “Solved and unsolved problems in the theory of trigonometric and orthogonal series”, Russian Math. Surveys, 19:1 (1964), 1–62  mathnet  crossref  mathscinet  zmath
16. G. G. Gevorkyan, “Uniqueness theorems for simple trigonometric series with application to multiple series”, Sb. Math., 212:12 (2021), 1675–1693  mathnet  crossref  crossref  mathscinet  zmath  adsnasa
17. N. Kholshchevnikova, V. Skvortsov, “On $U$- and $M$-sets for series with respect to characters of compact zero-dimensional groups”, J. Math. Anal. Appl., 446:1 (2017), 383–394  crossref  mathscinet  zmath
18. N. Kholshchevnikova, “The union problem and the category problem of sets of uniqueness in the theory of orthogonal series”, Real Anal. Exchange, 44:1 (2019), 65–76  crossref  mathscinet  zmath
19. S. F. Lukomskii, “On the uniqueness sets of multiple Walsh series for convergence in cubes”, Math. Notes, 109:3 (2021), 427–434  mathnet  crossref  crossref  mathscinet  zmath
20. G. G. Gevorkyan, “Uniqueness theorems for one-dimensional and double Franklin series”, Izv. Math., 84:5 (2020), 829–844  mathnet  crossref  crossref  mathscinet  zmath  adsnasa
21. G. G. Gevorkyan, L. A. Hakobyan, “Uniqueness theorems for multiple Franklin series converging over rectangles”, Math. Notes, 109:2 (2021), 208–217  mathnet  crossref  crossref  mathscinet  zmath
22. G. G. Gevorkyan, “Uniqueness theorems for Franklin series converging to integrable functions”, Sb. Math., 209:6 (2018), 802–822  mathnet  crossref  crossref  mathscinet  zmath  adsnasa
23. V. Skvortsov, “Recovering Banach-valued coefficients of series with respect to characters of zero-dimensional groups”, Ann. Univ. Sci. Budapest. Sect. Comput., 49 (2019), 379–397  mathscinet  zmath
24. M. Plotnikov, “$\mathcal V$-sets in the products of zero-dimensional compact abelian groups”, Eur. J. Math., 5:1 (2019), 223–240  crossref  mathscinet  zmath
25. V. A. Skvortsov, “Reconstruction of a generalized Fourier series from its sum on a compact zero-dimensional group in the non-abelian case”, Math. Notes, 109:4 (2021), 630–637  mathnet  crossref  crossref  mathscinet  zmath
26. J. M. Ash, Gang Wang, “Uniqueness questions for multiple trigonometric series”, Topics in harmonic analysis and ergodic theory, Contemp. Math., 444, Amer. Math. Soc., Providence, RI, 2007, 129–165  crossref  mathscinet  zmath
27. J. O. Smith III, Mathematics of the Discrete Fourier Transform (DFT), with audio applications, 2 ed., W3K Publishing, 2007, 306 pp.
28. I. W. Selesnick, G. Schuller, “The discrete Fourier transform”, The transform and data compression textbook, Ch. 2, CRC Press LLC, Boca Raton, FL, 2001, 37–79
29. N. M. Korobov, Exponential sums and their applications, Math. Appl. (Soviet Ser.), 80, Kluwer Acad. Publ., Dordrecht, 1992, xvi+209 pp.  crossref  mathscinet  mathscinet  zmath  zmath
30. L. Carleson, “On convergence and growth of partial sums of Fourier series”, Acta. Math., 116 (1966), 135–157  crossref  mathscinet  zmath
31. R. A. Hunt, “On the convergence of Fourier series”, Orthogonal expansions and their continuous analogues (Edwardsville, IL, 1967), Southern Illinois Univ. Press, Carbondale, IL, 1968, 235–255  mathscinet  zmath
32. P. Sjölin, “An inequality of Paley and convergence a.e. of Walsh–Fourier series”, Ark. Mat., 7:6 (1969), 551–570  crossref  mathscinet  zmath  adsnasa
33. N. Yu. Antonov, “Convergence of Fourier series”, East J. Approx., 2:2 (1996), 187–196  mathscinet  zmath
34. P. L. Ulyanov, “O ryadakh po perestavlennoi trigonometricheskoi sisteme”, Izv. AN SSSR. Ser. matem., 22:4 (1958), 515–542  mathnet  mathscinet  zmath
35. J. M. Ash, Sh. T. Tetunashvili, “Uniqueness for multiple trigonometric and Walsh series with convergent rearranged square partial sums”, Proc. Amer. Math. Soc., 134:6 (2006), 1681–1686  crossref  mathscinet  zmath
36. J. M. Ash, C. Freiling, D. Rinne, “Uniqueness of rectangularly convergent trigonometric series”, Ann. of Math. (2), 137:1 (1993), 145–166  crossref  mathscinet  zmath
37. N. N. Kholshchevnikova, “Union of sets of uniqueness for multiple Walsh and multiple trigonometric series”, Sb. Math., 193:4 (2002), 609–633  mathnet  crossref  crossref  mathscinet  zmath
38. Sh. T. Tetunashvili, “On some multiple function series and the solution of the uniqueness problem for Pringsheim convergence of multiple trigonometric series”, Sb. Math., 73:2 (1992), 517–534  mathnet  crossref  mathscinet  zmath  adsnasa
39. L. D. Gogoladze, “On the problem of reconstructing the coefficients of convergent multiple function series”, Izv. Math., 72:2 (2008), 283–290  mathnet  crossref  crossref  mathscinet  zmath
40. T. A. Zhereb'eva, “On a class of sets of uniqueness for double trigonometric series”, Math. Notes, 87:6 (2010), 811–820  mathnet  crossref  crossref  mathscinet  zmath
41. J. Bourgain, “Spherical summation and uniqueness of multiple trigonometric series”, Internat. Math. Res. Notices, 1996:3 (1996), 93–107  crossref  mathscinet  zmath


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