Geodesic
On the basis property of the system of exponentials and trigonometric systems of sine and cosine functions in weighted grand Lebesgue spaces
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (2024), pp. 15-25 
M. I. Ismailov; I. F. Aliyarova. On the basis property of the system of exponentials and trigonometric systems of sine and cosine functions in weighted grand Lebesgue spaces. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (2024), pp. 15-25. http://geodesic.mathdoc.fr/item/VMUMM_2024_2_a1/
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     title = {On the basis property of the system of exponentials and trigonometric systems of sine and cosine functions in weighted grand {Lebesgue} spaces},
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     pages = {15--25},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The paper is focused on the basis property of the system of exponentials and trigonometric systems of sine and cosine functions in a separable subspace of the weighted grand Lebesgue space generated by the shift operator. In this paper, with the help of the shift operator, a separable subspace $G_{p),\rho } (a,b)$ of the weighted space of the grand Lebesgue space $L_{p),\rho } (a,b)$ is defined. The density in $G_{p),\rho } (a,b)$ of the set $G_{0}^{\infty } ([a,b])$ of infinitely differentiable and finite on $[a,b]$ functions is studied. It is proved that if the weight function $\rho $ satisfies the Mackenhoupt condition, then the system of exponentials $\left\{e^{int} \right\}_{n\in Z} $ forms a basis in $G_{p),\rho } (-\pi ,\pi )$, and trigonometric systems of sine $\left\{\sin nt\right\}_{n\ge 1} $ and cosine $\left\{\cos nt\right\}_{n\ge 0} $ functions form bases in $G_{p),\rho } (0,\pi )$.

[1] Sharapudinov I.I., “Some problems of approximation theory in spaces $L_{p(x)} (E)$”, Anal. Math., 33:2 (2007), 135–153 | DOI | MR | Zbl

[2] Bilalov B.T., Guseynov Z.G., “Basicity of a system of exponents with a piece-wise linear phase in variable spaces”, Mediterr. J. Math., 9:3 (2012), 487–498 | DOI | MR | Zbl

[3] Bilalov B.T., “O bazisnosti vozmuschennoi sistemy eksponent v prostranstvakh tipa Morri”, Sib. matem. zhurn., 60:2 (2019), 323–350 | MR | Zbl

[4] Bilalov B.T., Guliyeva A.A., “On basicity of the perturbed systems of exponents in Morrey–Lebesgue space”, Inter. J. Math., 25 (2014), 1450054, 1–10 | DOI | MR

[5] Viner N., Peli R., Preobrazovanie Fure v kompleksnoi oblasti, Nauka, M., 1964

[6] Levin B.Ya., “O bazisakh pokazatelnykh funktsii v $L_{2} (-\pi ,\pi )$”, Zap. matem. otd. fiz.-mat. f-ta KhGU i KhMO. Ser. 4, 1961, no. 27, 39–48

[7] Pavlov B.S., “Bazisnost sistemy eksponent i uslovie Makenkhoupta”, Dokl. AN SSSR, 247 (1979), 37–40 | Zbl

[8] Nikolskii N.K., Pavlov B.S., Khruschev S.V., Bezuslovnye bazisy iz eksponent i vosproizvodyaschikh yader, Preprinty LOMI No 8-80, L., 1980

[9] Moiseev E.I., “O bazisnosti odnoi sistemy sinusov”, Differents. uravneniya, 23:1 (1987), 177–179 | MR | Zbl

[10] Sedletskii A.M., “O skhodimosti negarmonicheskikh ryadov Fure po sistemam eksponent, kosinusov i sinusov”, Dokl. AN SSSR, 301:5 (1988), 501–504

[11] Bilalov B.T., “Bazisnost nekotorykh sistem eksponent, kosinusov i sinusov”, Differents. uravneniya, 26:1 (1990), 10–16 | MR

[12] Moiseev E.I., “O bazisnosti sistem sinusov i kosinusov v vesovom prostranstve”, Differents. uravneniya, 34:1 (1998), 40–44 | MR | Zbl

[13] Pukhov S.S., Sedletskii A.M., “Bazisy iz eksponent, sinusov i kosinusov v vesovykh prostranstvakh na konechnom intervale”, Dokl. RAN, 425:4 (2009), 452–455 | Zbl

[14] Yukhimenko A.A., “Bazisy iz eksponent v vesovykh prostranstvakh $L_{2} (-\pi ,\pi )$”, Vestn. Mosk. un-ta. Matem. Mekhan., 2010, no. 2, 36–38 | MR

[15] Pukhov S.S., “Bazisy iz eksponent, sinusov i kosinusov v vesovykh prostranstvakh na konechnom intervale”, Izv. RAN. Ser. matem., 75:2 (2011), 195–224 | DOI | MR | Zbl

[16] Babenko K.I., “O sopryazhennykh funktsiyakh”, Dokl. AN SSSR, 62:2 (1948), 157–160 | Zbl

[17] Gaposhkin V.F., “Odno obobschenie teoremy M. Rissa o sopryazhennykh funktsiyakh”, Matem. sb., 46 (88):3 (1958), 111–115

[18] Hunt R.A., Muckenhoupt B., Wheeden R.L., “Weighed norm inequalities for the conjugated function and Hilbert transform”, Trans. Amer. Math. Soc., 176 (1973), 227–251 | DOI | MR | Zbl

[19] Hunt R.A., Young W.S., “A weighted norm inequality for Fourier series”, Bull. Amer. Math. Soc., 80 (1974), 274–277 | DOI | MR | Zbl

[20] Golubeva E.S., “Freimy eksponent so stepennym vesom”, Vestn. SamGU. Estestvennonauch. ser., 2011, no. 2(83), 15–25

[21] Bilalov B.T., Guliyeva A.A., “On the frame properties of degenerate system of sines”, J. Funct. Spaces Appl., 2012, 184186 | MR | Zbl

[22] Bilalov B.T., Mamedova Z.V., “On the frame properties of some degenerate trigonometric systems”, Dokl. Nats. Akad. Nauk Azerb., 68:5 (2012), 14–18 | MR | Zbl

[23] Shukyurov A.Sh., “O bazisnykh svoistvakh vzveshennykh eksponentsialnykh sistem s izbytkom”, Vestn. SamGU. Estestvennonauch. ser., 24:5 (2018), 14–19

[24] Shukurov A.Sh., “On basicity of the degenerate trigonometric system with excess”, Acta Comment. Univ. Tartu. Math., 21:2 (2017), 249–257 | MR | Zbl

[25] Kokilashvili V.M., Meskhi A., Rafeiro H., Samko S., “Integral Operators in Non-Standart Function Spaces”, Variable Exponent Holder, Morrey-Companato and Grand Spaces, v. 2, Birkhauser, Switzerland, 2016 | MR

[26] Dynkin E.M., Osilenker B.P., “Vesovye otsenki singulyarnykh integralov i ikh prilozheniya”, Itogi nauki i tekhniki. Ser. Matem. analiz, 21, 1983, 42–129