Geodesic
On Weyl multipliers of the rearranged trigonometric system
Sbornik. Mathematics, Tome 211 (2020) no. 12, pp. 1704-1736 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove that the condition $\sum_{n=1}^\infty1/(nw(n))<\infty$ is necessary for an increasing sequence of numbers $w(n)$ to be an almost everywhere unconditional convergence Weyl multiplier for the trigonometric system. This property was known long ago for Haar, Walsh, Franklin and some other classical orthogonal systems. The proof of this result is based on a new sharp logarithmic lower bound on $L^2$ for the majorant operator related to the rearranged trigonometric system. Bibliography: 32 titles.
Keywords: trigonometric series, Menshov-Rademacher theorem.
Mots-clés : Weyl multiplier 
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G. A. Karagulyan. On Weyl multipliers of the rearranged trigonometric system. Sbornik. Mathematics, Tome 211 (2020) no. 12, pp. 1704-1736. http://geodesic.mathdoc.fr/item/SM_2020_211_12_a1/

[1] S. V. Bočkarev, “Rearrangements of Fourier–Walsh series”, Math. USSR-Izv., 15:2 (1980), 259–275 | DOI | MR | Zbl

[2] S. V. Bočkarev, “On a majorant of the partial sums for a rearranged Walsh system”, Soviet Math. Dokl., 19:2 (1978), 341–342 | MR | Zbl

[3] C. Demeter, “Singular integrals along $N$ directions in $\mathbb R^2$”, Proc. Amer. Math. Soc., 138:12 (2010), 4433–4442 | DOI | MR | Zbl

[4] G. G. Gevorkyan, “Weyl multipliers for the unconditional convergence of series in a Franklin system”, Math. Notes, 41:6 (1987), 446–451 | DOI | MR | Zbl

[5] S. Sh. Galstyan, “On convergence and unconditional convergence of Fourier series”, Dokl. Math., 45:2 (1992), 286–289 | MR | Zbl

[6] G. A. Karagulyan, “On systems of non-overlapping Haar polynomials”, Ark. Mat., 58:1 (2020), 121–131 | DOI | Zbl

[7] S. Kaczmarz, H. Steinhaus, Theorie der Orthogonalreihen, Monogr. Mat., 6, Subwencji funduszu kultury narodowej, Warszawa–Lwow, 1935, vi+298 pp. | MR | Zbl

[8] B. S. Kashin, A. A. Saakyan, Orthogonal series, Transl. Math. Monogr., 75, Amer. Math. Soc., Providence, RI, 1989, xii+451 pp. | DOI | MR | MR | Zbl | Zbl

[9] A. Kolmogoroff, D. Menchoff, “Sur la convergence des séries de fonctions orthogonales”, Math. Z., 26:1 (1927), 432–441 | DOI | MR | Zbl

[10] D. E. Menchoff, “Sur les séries de fonctions orthogonales. I”, Fund. Math., 4 (1923), 82–105 | DOI | Zbl

[11] F. Móricz, “On the convergence of Fourier series in every arrangement of the terms”, Acta Sci. Math. (Szeged), 31 (1970), 33–41 | MR | Zbl

[12] S. Nakata, “On the divergence of rearranged Fourier series of square integrable functions”, Acta Sci. Math. (Szeged), 32 (1971), 59–70 | MR | Zbl

[13] S. Nakata, “On the divergence of rearranged trigonometric series”, Tohoku Math. J. (2), 27:2 (1975), 241–246 | DOI | MR | Zbl

[14] S. Nakata, “On the unconditional convergence of Walsh series”, Anal. Math., 5:3 (1979), 201–205 | DOI | MR | Zbl

[15] S. Nakata, “On the divergence of rearranged Walsh series”, Tohoku Math. J. (2), 24:2 (1972), 275–280 | DOI | MR | Zbl

[16] A. M. Olevskii, “Raskhodyaschiesya ryady Fure”, Izv. AN SSSR. Ser. matem., 27:2 (1963), 343–366 | MR | Zbl

[17] W. Orlicz, “Zur Theorie der Orthogonalreihen”, Bull. Intern. Acad. Sci. Polon. Cracovie, 1927, 81–115 | Zbl

[18] S. N. Poleščuk, “On the unconditional convergence of orthogonal series”, Anal. Math., 7:4 (1981), 265–275 | DOI | MR | Zbl

[19] H. Rademacher, “Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen”, Math. Ann., 87:1-2 (1922), 112–138 | DOI | MR | Zbl

[20] Constructive function theory {'} 81 (Varna, 1981), eds. B. Sendov, B. Boyanov, D. Vacov, R. Maleev, S. Markov, T. Boyanov, Publ. House Bulgar. Acad. Sci., Sofia, 1983, 598 pp. | MR | Zbl

[21] K. Tandori, “Über die orthogonalen Funktionen. X. (Unbedingte Konvergenz.)”, Acta Sci. Math. (Szeged), 23 (1962), 185–221 | MR | Zbl

[22] K. Tandori, “Über die Divergenz der Walshschen Reihen”, Acta Sci. Math. (Szeged), 27 (1966), 261–263 | MR | Zbl

[23] P. L. Ul'yanov, “Divergent Fourier series”, Russian Math. Surveys, 16:3 (1961), 1–75 | DOI | MR | Zbl

[24] P. L. Ul'yanov, “Divergent Fourier series of class $L^p$ ($p\ge2$)”, Soviet Math. Dokl., 2 (1961), 350–354 | MR | Zbl

[25] P. L. Ulyanov, “O mnozhitelyakh Veilya dlya bezuslovnoi skhodimosti”, Matem. sb., 60(102):1 (1963), 39–62 | MR | Zbl

[26] P. L. Ul'yanov, “On Weyl multipliers for unconditional convergence of orthogonal series”, Soviet Math. Dokl., 18 (1977), 1110–1113 | MR | Zbl

[27] P. L. Ul'yanov, “Exact Weil convergence factors for unconditional convergence”, Soviet Math. Dokl., 2 (1961), 1621–1622 | MR | Zbl

[28] 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

[29] P. L. Ul'yanov, “Kolmogorov and divergent Fourier series”, Russian Math. Surveys, 38:4 (1983), 57–100 | DOI | MR | Zbl

[30] P. L. Ul'yanov, “Works of D. E. Men'shov on the theory of orthogonal series and their further development”, Moscow Univ. Math. Bull., 47:4 (1992), 8–20 | MR | Zbl

[31] Z. Zahorski, “Une série de Fourier permutée d'une fonction de classe $L^{2}$ divergente presque partout”, C. R. Acad. Sci. Paris, 251 (1960), 501–503 | MR | Zbl

[32] A. Zygmund, Trigonometric series, v. II, 2nd ed., reprinted with corr. and some additions, Cambridge Univ. Press, London–New York, 1968, vii+364 pp. | MR | MR | Zbl | Zbl