Series in Multiplicative Systems in Lorentz Spaces

S. S. Volosivets

Matematičeskie zametki, Tome 102 (2017) no. 3, pp. 339-354.

Résumé

Series in multiplicative systems $\chi$ with generalized monotone coefficients are studied. Necessary and sufficient Hardy–Littlewood type conditions for the sums of such series to belong to the Lorentz space are proved. As corollaries, we establish estimates of best approximation in the system $\chi$ and Konyushkov-type theorems on the equivalence of $O$- and $\asymp$-relations for the weighted sums of the Fourier coefficients in the system $\chi$ and for the best approximations.

Détail

Keywords: Lorentz space, multiplicative systems, best approximation, Konyushkov-type theorems on the equivalence of $O$- and $\asymp$-relations.

Bibliographie

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

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

[3] G. G. Lorentz, “Some new functional spaces”, Ann. of Math. (2), 51 (1950), 37–55 | DOI | MR | Zbl

[4] N. K. Bari, S. B. Stechkin, “Nailuchshie priblizheniya i differentsialnye svoistva dvukh sopryazhennykh funktsii”, Tr. MMO, 5, GITTL, M., 1956, 483–522 | MR | Zbl

[5] A. A. Konyushkov, “Nailuchshie priblizheniya trigonometricheskimi polinomami i koeffitsienty Fure”, Matem. sb., 44 (86):1 (1958), 53–84 | MR | Zbl

[6] S. Tikhonov, “Trigonometric series with general monotone coefficients”, J. Math. Anal. Appl., 326:1 (2007), 721–735 | DOI | MR | Zbl

[7] M. Dyachenko, S. Tikhonov, “Integrability and continuity of functions represented by trigonometric series: coefficients criteria”, Stud. Math., 193:3 (2009), 285–306 | DOI | MR | Zbl

[8] G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli, A. I. Rubinshtein, Multiplikativnye sistemy funktsii i garmonicheskii analiz na nul-mernykh gruppakh, Elm, Baku, 1981 | MR | Zbl

[9] G. H. Hardy, J. E. Littlewood, “Notes on the theory of series (XIII): Some new properties of Fourier constants”, J. London Math. Soc., 6 (1931), 3–9 | DOI | MR | Zbl

[10] R. P. Boas, Jr., Integrability theorems for trigonometric transforms, Ergeb. Math. Grenzgeb., 38, Springer-Verlag, Berlin, 1967 | MR | Zbl

[11] M. F. Timan, A. I. Rubinshtein, “O vlozhenii klassov funktsii, opredelennykh na nulmernykh gruppakh”, Izv. vuzov. Matem., 1980, no. 8, 66–76 | MR | Zbl

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

[13] B. Booton, “General monotone sequences and trigonometric series”, Math. Nachr., 287:5-6 (2014), 518–529 | DOI | MR | Zbl

[14] B. V. Simonov, “O trigonometricheskikh ryadakh s monotonnymi koeffitsientami v prostranstvakh Lorentsa”, Izv. vuzov. Matem., 1998, no. 5, 59–67 | MR

[15] S. S. Volosivets, “On Hardy and Bellman transforms of series with respect to multiplicative systems in symmetric spaces”, Anal. Math., 35:2 (2009), 131–148 | DOI | MR | Zbl

[16] A. U. Bimendina, E. S. Smailov, “Koeffitsienty Fure–Praisa klassa GM i nailuchshie priblizheniya funktsii v prostranstve Lorentsa $L_{p\theta}[0,1)$, $1

+\infty$, $1\theta+\infty$”, Funktsionalnye prostranstva, teoriya priblizhenii, smezhnye razdely matematicheskogo analiza, Tr. MIAN, 293, MAIK, M., 2016, 83–104 | DOI | MR | Zbl

[17] L. Leindler, “Generalization of inequalities of Hardy and Littlewood”, Acta Sci. Math. (Szeged), 31:3-4 (1970), 279–285 | MR | Zbl

[18] L. Leindler, “Inequalities of Hardy–Littlewood type”, Anal. Math., 2:2 (1976), 117–123 | DOI | MR | Zbl

[19] G. G. Khardi, Dzh. E. Littlvud, G. Polia, Neravenstva, IL, M., 1948 | MR | Zbl

[20] B. Muckenhoupt, “Hardy's inequality with weights”, Stud. Math., 44:1 (1972), 31–38 | MR | Zbl

[21] N. Yu. Agafonova, “O nailuchshikh priblizheniyakh funktsii po multiplikativnym sistemam i svoistvakh ikh koeffitsientov Fure”, Anal. Math., 33:4 (2007), 247–262 | DOI | MR | Zbl