Geodesic
Fourier–Price coefficients of class GM and best approximations of functions in the Lorentz space $L_{p\theta}[0,1)$, $1$, $1\theta+\infty$
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximation theory, and related problems of mathematical analysis, Tome 293 (2016), pp. 83-104 Cet article a éte moissonné depuis la source Math-Net.Ru

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For polynomials in the Price system, we establish an inequality of different metrics in the Lorentz spaces. Applying this inequality, we prove a Hardy–Littlewood theorem for the Fourier–Price series with GM sequences of coefficients in the two-parameter Lorentz spaces and in the Nikol'skii–Besov spaces with a Price basis. We also study the behavior of the best approximations of functions by Price polynomials in the metric of the Lorentz space. 
@article{TM_2016_293_a5,
     author = {A. U. Bimendina and E. S. Smailov},
     title = {Fourier{\textendash}Price coefficients of class {GM} and best approximations of functions in the {Lorentz} space $L_{p\theta}[0,1)$, $1<p<+\infty$, $1<\theta<+\infty$},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {83--104},
     year = {2016},
     volume = {293},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2016_293_a5/}
}
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A. U. Bimendina; E. S. Smailov. Fourier–Price coefficients of class GM and best approximations of functions in the Lorentz space $L_{p\theta}[0,1)$, $1

[1] Agafonova N.Yu., “O nailuchshikh priblizheniyakh funktsii po multiplikativnym sistemam i svoistvakh ikh koeffitsientov Fure”, Anal. math., 33 (2007), 247–262 | DOI | MR | Zbl

[2] Berg I., Lëfstrëm I., Interpolyatsionnye prostranstva. Vvedenie, Mir, M., 1980 | MR

[3] Bokaev N.A., Smailov E.S., Ryady Fure po multiplikativnym sistemam, Ucheb. posobie, KarGU, Karaganda, 1993

[4] Borisova E.A., O vlozhenii klassov funktsii, zadannykh posledovatelnostyami nailuchshikh priblizhenii po nekotorym ortonormirovannym sistemam, diss. ... kand. fiz.-mat. nauk, MGU, M., 1988

[5] Dyachenko M.I., “Teorema Khardi–Littlvuda dlya trigonometricheskikh ryadov s obobschenno-monotonnymi koeffitsientami”, Izv. vuzov. Matematika, 2008, no. 5, 38–47 | MR | Zbl

[6] Dyachenko M.I., Nursultanov E.D., “Teorema Khardi–Littlvuda dlya trigonometricheskikh ryadov s $\alpha $-monotonnymi koeffitsientami”, Mat. sb., 200:11 (2009), 45–60 | DOI | MR | Zbl

[7] Golubov B.I., Efimov A.V., Skvortsov V.A., Ryady i preobrazovaniya Uolsha, Nauka, M., 1987 | MR

[8] Goldman M.L., “O vlozhenii prostranstv Nikolskogo–Besova s modulyami nepreryvnosti obschego vida v prostranstva Lorentsa”, DAN SSSR, 277:1 (1984), 20–24 | MR | Zbl

[9] Gulisashvili A.B., “Trigonometricheskie ryady s monotonno ubyvayuschimi koeffitsientami i funktsii raspredeleniya”, Mat. zametki, 10:1 (1971), 3–10 | Zbl

[10] Gulisashvili A.B., Rodin V.A., Semenov E.M., “Koeffitsienty Fure summiruemykh funktsii”, Mat. sb., 102:3 (1977), 362–371 | MR | Zbl

[11] Khardi G.G., Littlvud Dzh.E., Polia G., Neravenstva, Izd-vo inostr. lit., M., 1948

[12] Kokilashvili V.M., “O priblizhenii periodicheskikh funktsii”, Tr. Tbil. mat. in-ta, 34 (1968), 51–81 | Zbl

[13] Konyushkov A.A., “Nailuchshie priblizheniya trigonometricheskimi polinomami i koeffitsienty Fure”, Mat. sb., 44:1 (1958), 53–84

[14] Krein S.G., Petunin Yu.I., Semenov E.M., Interpolyatsiya lineinykh operatorov, Nauka, M., 1978 | MR

[15] Leindler L., “Best approximation and Fourier coefficients”, Anal. math., 31 (2005), 117–129 | DOI | MR | Zbl

[16] Nursultanov E.D., “Setevye prostranstva i neravenstva tipa Khardi–Littlvuda”, Mat. sb., 189:3 (1998), 83–102 | DOI | MR | Zbl

[17] Nursultanov E.D., “O koeffitsientakh kratnykh ryadov Fure iz $L_p$-prostranstv”, Izv. RAN. Ser. mat., 64:1 (2000), 95–122 | DOI | MR | Zbl

[18] Rodin V.A., “Teorema Khardi–Littlvuda dlya sinus-ryada v simmetrichnom prostranstve”, Mat. zametki, 20:2 (1976), 241–246 | MR | Zbl

[19] Rodin V.A., “O prinadlezhnosti summy kosinus-ryada s monotonnymi koeffitsientami simmetrichnomu prostranstvu”, Izv. vuzov. Matematika, 1979, no. 8, 60–64 | MR

[20] Sagher Y., “An application of interpolation theory to Fourier series”, Stud. math., 41:2 (1972), 169–181 | MR | Zbl

[21] Semenov E.M., “Interpolyatsiya lineinykh operatorov i otsenki koeffitsientov Fure”, DAN SSSR, 176:6 (1967), 1251–1254 | Zbl

[22] Smailov E.S., Bimendina A.U., “Teorema Khardi–Littlvuda dlya ryadov Fure–Praisa s kvazimonotonnymi koeffitsientami v prostranstve Lorentsa”, Vestn. Karagand. un-ta. Matematika, 2005, no. 2, 3–9

[23] Smailov E.S., Bimendina A.U., “O vlozhenii v prostranstvo Lorentsa”, Vestn. Karagand. un-ta. Matematika, 2008, no. 2, 75–83

[24] Smailov E.S., Suleimenova Z.R., “Teoremy vlozheniya dlya prostranstv Besova po multiplikativnym bazisam Praisa”, Tr. MIAN, 243 (2003), 313–319 | MR | Zbl

[25] Stein I., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974

[26] Tazabekov S., Smailov E.S., Trigonometricheskie ryady Fure s kvazimonotonnymi koeffitsientami, Dep. v VINITI, No5255-V88, M., 1988

[27] Tikhonov S., “Trigonometric series with general monotone coefficients”, Math. Anal. Appl., 326 (2007), 721–735 | DOI | MR | Zbl

[28] Timan M.F., Tukhliev K., “Svoistva nekotorykh ortonormirovannykh sistem”, Izv. vuzov. Matematika, 1983, no. 9, 65–73 | MR | Zbl

[29] Ulyanov P.L., “Teoremy vlozheniya i sootnosheniya mezhdu nailuchshimi priblizheniyami (modulyami nepreryvnosti) v raznykh metrikakh”, Mat. sb., 81:1 (1970), 104–131 | Zbl

[30] Yong W.-S., “Mean convergence of generalized Walsh–Fourier series”, Trans. Amer. Math. Sos., 218 (1976), 311–320 | DOI | MR

[31] Zigmund A., Trigonometricheskie ryady, T. 2, Mir, M., 1965 | MR