Transformation of Fourier Series by Means of General Monotone Sequences
Matematičeskie zametki, Tome 107 (2020) no. 5, pp. 674-692.

Voir la notice de l'article provenant de la source Math-Net.Ru

Estimates of the norms and the best approximations of the generalized Liouville–Weyl derivative in the two-dimensional case are obtained.
Keywords: best approximation, best approximation by a two-dimensional angle.
@article{MZM_2020_107_5_a2,
     author = {A. A. Jumabayeva and B. V. Simonov},
     title = {Transformation of {Fourier} {Series} by {Means} of {General} {Monotone} {Sequences}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {674--692},
     publisher = {mathdoc},
     volume = {107},
     number = {5},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2020_107_5_a2/}
}
TY  - JOUR
AU  - A. A. Jumabayeva
AU  - B. V. Simonov
TI  - Transformation of Fourier Series by Means of General Monotone Sequences
JO  - Matematičeskie zametki
PY  - 2020
SP  - 674
EP  - 692
VL  - 107
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2020_107_5_a2/
LA  - ru
ID  - MZM_2020_107_5_a2
ER  - 
%0 Journal Article
%A A. A. Jumabayeva
%A B. V. Simonov
%T Transformation of Fourier Series by Means of General Monotone Sequences
%J Matematičeskie zametki
%D 2020
%P 674-692
%V 107
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2020_107_5_a2/
%G ru
%F MZM_2020_107_5_a2
A. A. Jumabayeva; B. V. Simonov. Transformation of Fourier Series by Means of General Monotone Sequences. Matematičeskie zametki, Tome 107 (2020) no. 5, pp. 674-692. http://geodesic.mathdoc.fr/item/MZM_2020_107_5_a2/

[1] B. V. Simonov, S. Yu. Tikhonov, “Teoremy vlozheniya v konstruktivnoi teorii priblizhenii”, Matem. sb., 199:9 (2008), 107–148 | DOI | MR | Zbl

[2] S. Bernstein, “On the best approximation of continuous functions by polynomials of a given degree”, Comm. Soc. Math. Kharkow, 13:2 (1912), 49–194

[3] R. DeVore, G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin, 1993 | MR

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

[5] M. F. Timan, “Obratnye teoremy konstruktivnoi teorii funktsii v prostranstvakh $L_p$ ($1\le p\le\infty$)”, Matem. sb., 46 (88):1 (1958), 125–132 | MR | Zbl

[6] J. Marcinkiewicz, “Sur quelques intégrales du type de Dini”, Ann. Soc. Polon. Math., 17 (1938), 42–50 | Zbl

[7] O. V. Besov, “O nekotorykh usloviyakh prinadlezhnosti k $L_p$ proizvodnykh periodicheskikh funktsii”, Nauchn. dokl. vyssh. shkoly. Ser. fiz.-matem. nauki, 1 (1959), 13–17 | Zbl

[8] S. B. Stechkin, “O nailuchshem priblizhenii sopryazhennykh funktsii trigonometricheskimi polinomami”, Izv. AN SSSR. Ser. matem., 20:2 (1956), 197–206 | MR | Zbl

[9] M. F. Timan, “O vlozhenii $L_p^{(k)}$ klassov funktsii”, Izv. vuzov. Matem., 1974, no. 10, 61–74 | MR | Zbl

[10] B. V. Simonov, S. Tikhonov, “On embeddings of functional classes defined by constructive characteristics”, Approximation and Probability, Banach Center Publ., 72, Polish Acad. Sci. Inst. Math., Warsaw, 2006, 285–307 | DOI | MR

[11] I. Szalay, “On the best approximation of factorized Fourier series”, Approximation Theory, Reidel, Dordrecht, 1975, 235–241 | MR

[12] A. I. Stepanets, E. I. Zhukina, “Inverse theorems for the approximation of $(\psi,\beta)$-differentiable functions”, Ukr. Math. J., 41:8 (1989), 953–959 | DOI | MR

[13] A. Jumabayeva, “Liouville–Weyl derivatives, best approximations, and moduli of smoothness”, Acta Math. Hungar., 145:2 (2015), 369–391 | DOI | MR

[14] S. Tikhonov, “Embedding results in questions of strong approximation by Fourier series”, Acta Sci. Math. (Szeged), 72 (2006), 117–128 | MR

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

[16] M. Dyachenko, S. Tikhonov, “Convergence of trigonometric series with general monotone coefficients”, C. R. Math. Acad. Sci. Paris, 345:3 (2007), 123–126 | DOI | MR

[17] M. Dyachenko, S. Tikhonov, “A Hardy–Littlewood theorem for multiple series”, J. Math. Anal. Appl., 339:1 (2008), 503–510 | DOI | MR

[18] D. Gorbachev, E. Liflyand, S. Tikhonov, “Weighted Fourier inequalities: Boas' conjecture in $\mathbb R^n$”, J. Anal. Math., 114 (2011), 99–120 | DOI | MR

[19] S. Tikhonov, “On $L_1$-convergence of Fourier series”, J. Math. Anal. Appl., 347:2 (2008), 416–427 | DOI | MR

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

[21] S. Tikhonov, “On generalized Lipschitz classes and Fourier series”, Z. Anal. Anwendungen, 23:4 (2004), 745–764 | DOI | MR

[22] S. Yu. Tikhonov, “Obobschennye klassy Lipshitsa i koeffitsienty Fure”, Matem. zametki, 75:6 (2004), 947–951 | DOI | MR | Zbl

[23] E. Liflyand, S. Tikhonov, “A concept of general monotonicity and application”, Math. Nachr., 284:8-9 (2011), 1083–1098 | DOI | MR

[24] M. K. Potapov, B. V. Simonov, S. Yu. Tikhonov, Drobnye moduli gladkosti, Maks-Press, M., 2016

[25] M. K. Potapov, “K voprosu ob ekvivalentnosti uslovii skhodimosti ryadov Fure”, Matem. sb., 68 (110):1 (1965), 111–127 | MR | Zbl

[26] S. M. Nikolskii, Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1977 | MR | Zbl

[27] A. Zigmund, Trigonometricheskie ryady, Mir, M., 1965 | MR | Zbl | Zbl

[28] M. K. Potapov, “O priblizhenii uglom”, Proceedings of the Conference on the Constructive Theory of Functions (Budapest, 1969), Akadémiai Kiadó, Budapest, 1972, 371–399 | MR

[29] O. V. Besov, V. P. Ilin, S. M. Nikolskii, Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 | MR | Zbl

[30] A. Jumabayeva, B. Simonov, “Liouville–Weyl derivatives of double trigonometric series”, Appl. Numer. Harmonic Anal. (to appear)

[31] M. K. Potapov, B. V. Simonov, S. Yu. Tikhonov, “Sootnosheniya mezhdu smeshannymi modulyami gladkosti i teoremy vlozheniya klassov Nikolskogo”, Teoriya funktsii i differentsialnye uravneniya, Tr. MIAN, 269, MAIK «Nauka/Interperiodika», M., 2010, 204–214 | MR | Zbl

[32] B. V. Simonov, O prinadlezhnosti preobrazovannogo ryada Fure prostranstvu $L_p$, Dep. v VINITI No 4985-81, 1981

[33] M. G. Esmaganbetov, Usloviya suschestvovaniya smeshannykh proizvodnykh Veilya v $L_p([0,2\pi^2])$ ($1 p \infty$) i ee strukturnye svoistva, Dep. v VINITI No 1675-82, 1982

[34] M. K. Potapov, “Izuchenie nekotorykh klassov funktsii pri pomoschi priblizheniya “uglom””, Issledovaniya po teorii differentsiruemykh funktsii mnogikh peremennykh i ee prilozheniyam. IV, Tr. MIAN SSSR, 117, 1972, 256–291 | MR | Zbl