Geodesic
Estimates of functionals by the second moduli of continuity of even derivatives
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 41, Tome 416 (2013), pp. 70-90 Cet article a éte moissonné depuis la source Math-Net.Ru

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We establish an expansion of a function in terms of the second order differences of its derivatives. This expansion generalizes the well-known expansion in terms of the first order differences. Then, with the help of this expansion, we estimate some functionals by the second moduli of continuity. As particular cases of the estimates obtained, we have Jackson-type inequalities for approximations by entire fuctions of exponential type, trigonometric polynomials and splines in various function spaces. The constants in the new inequalities are smaller than those known before. 
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O. L. Vinogradov; V. V. Zhuk. Estimates of functionals by the second moduli of continuity of even derivatives. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 41, Tome 416 (2013), pp. 70-90. http://geodesic.mathdoc.fr/item/ZNSL_2013_416_a2/

[1] V. V. Zhuk, Approksimatsiya periodicheskikh funktsii, L., 1982 | MR

[2] V. V. Zhuk, “O nekotorykh tochnykh neravenstvakh mezhdu nailuchshimi priblizheniyami i modulyami nepreryvnosti”, Sib. mat. zhurn., 12:6 (1971), 1283–1297 | MR

[3] A. A. Ligun, “O tochnykh konstantakh priblizheniya differentsiruemykh periodicheskikh funktsii”, Mat. zametki, 14:1 (1973), 21–30 | MR | Zbl

[4] N. P. Korneichuk, Tochnye konstanty v teorii priblizheniya, M., 1987 | MR

[5] A. Yu. Gromov, “O tochnykh konstantakh priblizhenii tselymi funktsiyami differentsiruemykh funktsii”, Issledovaniya po sovremennym problemam summirovaniya i priblizheniya funktsii i ikh prilozheniyam, 7, Dnepropetrovsk, 1976, 17–21 | MR

[6] N. I. Akhiezer, Lektsii po teorii approksimatsii, M., 1965

[7] O. L. Vinogradov, “Tochnye neravenstva tipa Dzheksona dlya priblizhenii klassov svertok tselymi funktsiyami konechnoi stepeni”, Algebra i analiz, 17:4 (2005), 59–114 | MR | Zbl

[8] O. L. Vinogradov, V. V. Zhuk, “Tochnye neravenstva tipa Kolmogorova dlya modulei nepreryvnosti i nailuchshikh priblizhenii trigonometricheskimi mnogochlenami i splainami”, Zap. nauchn. semin. POMI, 290, 2002, 5–26 | MR | Zbl

[9] O. L. Vinogradov, “Analog summ Akhiezera–Kreina–Favara dlya periodicheskikh splainov minimalnogo defekta”, Problemy mat. analiza, 25, 2003, 29–56 | Zbl

[10] M. G. Krein, “O nailuchshei approksimatsii nepreryvnykh differentsiruemykh funktsii na vsei veschestvennoi osi”, Dokl. AN SSSR, 18:9 (1938), 619–623 | Zbl

[11] O. L. Vinogradov, “Tochnye neravenstva dlya priblizhenii klassov periodicheskikh svertok podprostranstvami sdvigov nechetnoi razmernosti”, Mat. zametki, 85:4 (2009), 569–584 | DOI | MR | Zbl

[12] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, v. 1, M., 1965

[13] O. L. Vinogradov, V. V. Zhuk, “Otsenki funktsionalov s izvestnym konechnym naborom momentov cherez moduli nepreryvnosti i povedenie konstant v neravenstvakh tipa Dzheksona”, Algebra i analiz, 24:5 (2012), 1–43 | MR

[14] O. L. Vinogradov, V. V. Zhuk, “Tochnye otsenki otklonenii lineinykh metodov priblizheniya periodicheskikh funktsii posredstvom lineinykh kombinatsii modulei nepreryvnosti razlichnykh poryadkov”, Problemy mat. analiza, 25, 2003, 57–98

[15] V. V. Zhuk, G. I. Natanson, “O konstantakh v pryamykh teoremakh teorii priblizheniya”, Vestn. Leningr. un-ta, ser. mat., mekh. astr., 1980, no. 7, 5–9 | MR | Zbl

[16] V. V. Zhuk, “O tochnosti predstavleniya nepreryvnoi $2\pi$-periodicheskoi funktsii pri pomoschi lineinykh metodov approksimatsii”, Izv. VUZov. Matematika, 1972, no. 8(123), 46–59 | MR | Zbl