Sharp inequalities of Jackson-Stechkin type for periodic functions in $L_2$ differentiable in the Weyl sense

M. Sh. Shabozov ; A. A. Shabozova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 4, pp. 255-264

Résumé

For periodic functions differentiable in the sense of Weyl and belonging to the space $L_{2}$, sharp inequalities of Jackson–Stechkin type are obtained for a special $m$th-order modulus of continuity generated by the Steklov operator (function). Similar characteristics of smoothness of functions were considered earlier by V. A. Abilov, F. V. Abilova, V. M. Kokilashvili, S. B. Vakarchuk, V. I. Zabutnaya, K. Tukhliev, etc. For classes of functions defined in terms of these characteristics, we solve a number of extremal problems of polynomial approximation theory.

Détail

Keywords: best approximation, periodic function, special modulus of continuity, Jackson–Stechkin inequalities, extremal problems.

Bibliographie

[1] Ditzian Z., Totik V., Moduli of smoothness, Springer Ser. Comput. Math., 9, Springer-Verlag, N Y, 1987, 227 pp. | DOI | MR | Zbl

[2] Runovskii K.V., “O priblizhenii semeistvami lineinykh polinomialnykh operatorov v prostranstvakh $L_{p}, 0 p 1$”, Mat. sb., 185:8 (1994), 81–102

[3] Vasilev S.N., “Tochnoe neravenstvo Dzheksona - Stechkina v $L_{2}$ s modulem nepreryvnosti, porozhdennymi proizvolnym konechno-raznostnym operatorom s postoyannymi koeffitsientami”, Dokl. RAN, 385:1 (2002), 11–14 | MR | Zbl

[4] Kozko A.I., Rozhdestvenskii A.V., “O neravenstve Dzheksona s obobschennym modulem nepreryvnosti”, Mat. zametki, 73:5 (2003), 783–788 | DOI | MR | Zbl

[5] Vakarchuk S.B., “Tochnye konstanty v neravenstvakh tipa Dzheksona i tochnye znacheniya poperechnikov funktsionalnykh klassov iz $L_2$”, Mat. zametki, 78:5 (2005), 792–796 | DOI | MR | Zbl

[6] Ivanov A.V., Ivanov V.I., “Optimalnye argumenty v neravenstve Dzheksona v prostranstve $L_2(\mathbb{R}^{d})$ so stepennym vesom”, Mat. zametki, 94:3 (2013), 338–348 | DOI | Zbl

[7] Potapov M.K., “O primenenii odnogo operatora obobschennogo sdviga v teorii priblizhenii”, Vest. Mosk. un-ta. Ser. 1. Matematika. Mekhanika, 1998, no. 3, 38–48 | Zbl

[8] Potapov M.K., “O primenenii nesimmetrichnykh operatorov obobschennogo sdviga v teorii priblizhenii”, Tr. matematicheskogo tsentra im. N.I. Lobachevskogo, 8 (2001), 185–189, Kazan

[9] Potapov M.K., “O svoistvakh i o primenenii v teorii priblizhenii odnogo svoistva operatorov obobschennogo sdviga”, Mat. zametki, 69:3 (2001), 412–426 | DOI | MR | Zbl

[10] Napadenina A.Yu., “O sovpadenii klassov funktsii, opredelyaemykh operatorami obobschennogo sdviga ili poryadkom nailuchshego priblizheniya”, Vestnik Mosk. un-ta. Ser. 1. Matematika. Mekhanika, 2004, no. 2, 29–33

[11] Abilov V.A., Abilova F.V., “Nekotorye voprosy priblizheniya $2\pi$-periodicheskikh funktsii summami Fure v prostranstve $L_{2}(2\pi)$”, Mat. zametki, 76:6 (2004), 803–811 | DOI | Zbl

[12] Kokilashvili V., Yildirir Y.E., “On the approximation in weighted Lebesgue spaces”, Proc. A. Ramzadze Math. Inst., 143 (2007), 103–113 | MR | Zbl

[13] Vakarchuk S.B., Zabutnaya V.I., “Neravenstva tipa Dzheksona - Stechkina dlya spetsialnykh modulei nepreryvnosti i poperechniki funktsionalnykh klassov v prostranstve $L_2$”, Mat. zametki, 92:4 (2012), 497–514 | DOI | Zbl

[14] Shabozov M.Sh., Tukhliev K., “Nailuchshie polinomialnye priblizheniya i poperechniki nekotorykh funktsionalnykh klassov v $L_{2}$”, Mat. zametki, 94:6 (2013), 908–917 | DOI | Zbl

[15] Weyl H., “Bemerkungen zum Begriff der differential quotienten gebrochener Ordnung”, Vierteljahresschr. Natursch. Ges. Zurich, 62 (1917), 296–302 | MR | Zbl

[16] Taikov L.V., “Neravenstva, soderzhaschie nailuchshie priblizheniya i modul nepreryvnosti funktsii iz $L_{2}$”, Mat. zametki, 20:3 (1976), 433–438 | MR | Zbl

[17] Shabozov M.Sh., Yusupov G.A., “Nailuchshie polinomialnye priblizheniya v $L_{2}$ nekotorykh klassov $2\pi$-periodicheskikh funktsii i tochnye znacheniya ikh poperechnikov”, Mat. zametki, 90:5 (2011), 764–775 | DOI | MR | Zbl