Geodesic
On unimprovability of some theorems on convergence in mean of trigonometric series
Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2018) no. 1, pp. 31-49.

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The paper puts forward a method for constructing trigonometric Fourier series with $L_{2\pi}$-unbounded partial sums that have coefficients with some preassigned properties. In particular, examples of trigonometric Fourier series showing that some conditions of convergence in the mean of trigonometric series cannot be sharpened are constructed. 
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A. S. Belov. On unimprovability of some theorems on convergence in mean of trigonometric series. Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2018) no. 1, pp. 31-49. http://geodesic.mathdoc.fr/item/FPM_2018_22_1_a2/

[1] Bari N. K., Trigonometricheskie ryady, Fizmatgiz, M., 1961

[2] Belov A. S., “Ob usloviyakh skhodimosti v srednem trigonometricheskikh ryadov Fure”, Izv. Tul. gos. un-ta. Ser. Matematika. Mekhanika. Informatika, 4:1 (1998), 40–46 | MR

[3] Belov A. S., “Ob usloviyakh skhodimosti (ogranichennosti) v srednem chastnykh summ trigonometricheskogo ryada”, Metricheskaya teoriya funktsii i smezhnye voprosy analiza, Sb. statei, AFTs, M., 1999, 1–17

[4] Belov A. S., “O koeffitsientnykh usloviyakh skhodimosti trigonometricheskogo ryada v srednem”, Teoriya funktsii, ee prilozheniya i smezhnye voprosy, Materialy 5-i Kazanskoi mezhdunarodnoi shkoly-konferentsii (27 iyunya–4 iyulya 2001 g.), Tr. matem. tsentra im. N. I. Lobachevskogo, 8, Izd-vo DAS, Kazan, 2001, 36–38

[5] Belov A. S., “Ob odnom uslovii skhodimosti v srednem trigonometricheskikh ryadov”, Matem. zametki, 69:3 (2001), 323–328 | DOI | MR | Zbl

[6] Belov A. S., “Zamechaniya o skhodimosti (ogranichennosti) v srednem chastnykh summ trigonometricheskogo ryada”, Matem. zametki, 71:6 (2002), 807–817 | DOI | Zbl

[7] Belov A. S., “O novykh usloviyakh skhodimosti v srednem trigonometricheskikh ryadov”, Tr. mezhdunar. shkoly-seminara po geometrii i analizu pamyati N. V. Efimova, Tezisy dokladov (Abrau-Dyurso, baza otdykha Rostovskogo gosuniv. «Limanchik», 5–11 sent. 2002 g.), Rostov-na-Donu, 2002, 103–105

[8] Belov A. S., “Ob usloviyakh skhodimosti (ogranichennosti) v srednem chastnykh summ trigonometricheskogo ryada”, Vestn. Ivanov. gosuniv. Ser. Biologiya. Khimiya. Fizika. Matematika, 2004, no. 3, 110–117

[9] Belov A. S., “Neuluchshaemost nekotorykh teorem o skhodimosti v srednem trigonometricheskikh ryadov Fure”, Materialy 9-i mezhdunar. Kazanskoi letnei nauch. shkoly-konferentsii «Teoriya funktsii, ee prilozheniya i smezhnye voprosy» (Kazan, 1–7 iyulya 2009 g.), Tr. matem. tsentra im. N. I. Lobachevskogo, 38, Kazan. matem. obsch-vo, Kazan, 2009, 40–43

[10] Zigmund A., Trigonometricheskie ryady, v. 1, 2, Mir, M., 1965 | MR

[11] Telyakovskii S. A., Fomin G. A., “O skhodimosti v metrike $L$ ryadov Fure s kvazimonotonnymi koeffitsientami”, Tr. MIAN SSSR, 134, 1975, 310–313

[12] Fomin G. A., “O skhodimosti ryadov Fure v srednem”, Matem. sb., 110:2 (1979), 251–265 | MR | Zbl

[13] Fridli S., “On the $L^1$-convergence of Fourier series”, Stud. Math., 125:2 (1997), 161–174 | DOI | MR | Zbl

[14] Garret J. W., Rees C. S., Stanojević Č. V., “On $L^1$-convergence of Fourier series with quasi-monotone coefficients”, Proc. Amer. Math. Soc., 72:3 (1978), 535–538 | MR | Zbl