Entire functions of order 1/2 in the approximation of functions on the semiaxis
DOI:
https://doi.org/10.21638/11701/spbu01.2019.408Abstract
Let In be disjoint segments of half-axis [1;1), I1 = [1; b1]. We assume that In and their supplementary intervals are commensurable. The union of mentioned segments we denote thought E. Let 0 < α< 1. The class of complex-valued functions defined on E with r-th derivative belonging to a Holder class α and bounded by 1 is denoted by H_1^(r+α)(E). We consider the set Cσ of entire functions of the order 1/2 and of the type ≤ σ which are bounded on [0;1). The main result of this paper is the following: Let f ∈ 2 H_1^(r+α)(E). Then there exist constants c0 and c1, depending of f, E, r and α such that for any σ ≥ 1 one can find a function Fσ ∈ Cσ such that the following inequality holds |f(x) − Fσ(x)|≤ min(c0∙1/σr+α(√((x-a_n)(b_n )-x)+n/σ)r+α,c1), x ∈ In, n ≥ 1.
Keywords:
smooth functions, entire functions, approximation
Downloads
Download data is not yet available.
References
Литература
Давыдова Т.С., Широков Н.А. Приближение функций из класса Гёльдера на полуоси // Записки научн. семинаров ПОМИ. 1999. Т. 262. С. 127–137.
Сильванович О.В., Широков Н.А. Приближение целыми функциями на подмножествах полуоси // Записки научн. семинаров ПОМИ. 2006. Т. 337. С. 233–237.
Silvanovich O.V., Shirokov N.A. Approximation by entire functions on a countable union of segments on the real axis. 1. Formulation of the results // Vestn. St. Petersburg Univ.: Math. 2016. Vol. 49. Issue 4. P. 373–376. https://doi.org/10.3103/S1063454116040130
Silvanovich O.V., Shirokov N.A. Approximation by entire functions on a countable union of segments on the real axis. 2. Proof of the Main Theorem // Vestn. St. Petersburg Univ.: Math. 2017. Vol. 50. Issue 1. P. 35–43. https://doi.org/10.3103/S1063454117010125
Silvanovich O.V., Shirokov N.A. Approximation by entire functions on a countable union of segments on the real axis. 3. Future Generalization // Vestn. St. Petersburg Univ.: Math. 2018. Vol. 51. Issue 2. P. 164–168. https://doi.org/10.3103/S1063454118020085
Бабенко В.Ф. Экстремальные задачи теории приближения и неравенства для перестановок // Доклады АН СССР. 1986. Т. 290, №5. С. 1033–1036.
Бабенко В.Ф. Точные неравенства для норм промежуточных производных полуцелого порядка и некоторые их приложения // Доп. НАН Украiны. 1995. №2. С. 23–26.
Бабенко В.Ф. О неравенствах для норм промежуточных производных на конечном интервале // Укр. мат. журнал. 1995. Т. 47, №1. С. 105–107.
Бабенко В.Ф., Кофанов В.А., Пичугов С.А. Об аддитивных неравенствах для норм промежуточных производных // Доклады РАН. 1997. Т. 356, №2. С. 154–156.
Буренков В.И. О точных постоянных в неравенствах для норм промежуточных производных на конечном интервале // Труды Мат. института им. В. А. Стеклова АН СССР. 1980. Vol. 156. С. 22–29.
Кофанов В.А. О неравенствах типа Ландау — Колмогорова — Хермандера на отрезке и вещественной прямой // Укр. мат. журнал. 2000. Т. 52, №12. С. 1676–1688.
Chen W. Landau — Kolmogorov inequality on a finite interval // Bull. Austral. Math. Soc. 1993. Vol. 48. P. 485–494. https://doi.org/10.1017/S000497270001594X
Шадрин А.Ю. О точных постоянных в неравенствах между L1-нормами производных на конечном отрезке // Доклады РАН. 1992. Т. 326, №1. С. 150–153.
Dyn’kin E.M. Pseudoanalytic extensions of smooth functions. The uniform scale // Amer. Math. Soc. Transl. 1980. Vol. 115. P. 33–38. https://doi.org/10.1090/trans2/115/02
Dyn’kin E.M. The pseudoanalytic extensions // J. Anal. Math. 1993. Vol. 60. P. 45–70. https://doi.org/10.1007/BF03341966
References
Davidova T.S., Shirokov N.A., “Approximation of functions from class Gelder on the semiaxis”, Zapiski nauch. sem. POMI 262, 127–137 (1999). (In Russian)
Silvanovich O.V., Shirokov N.A., “Approximation by entire functions on subsets of the semiaxis”, Zapiski nauch. sem. POMI 337, 233–237 (2006). (In Russian)
Silvanovich O.V., Shirokov N.A., “Approximation by entire functions on a countable union of segments on the real axis. 1. Formulation of the results”, Vestn. St.Petersburg Univ.: Math. 49, issue 4, 373–376 (2016). https://doi.org/10.3103/S1063454116040130
Silvanovich O.V., Shirokov N.A., “Approximation by entire functions on a countable union of segments on the real axis. 2. Proof of the Main Theorem”, Vestn. St. Petersburg Univ.: Math. 50, issue 1, 35–43 (2017). https://doi.org/10.3103/S1063454117010125
Silvanovich O.V., Shirokov N.A., “Approximation by entire functions on a countable union of segments on the real axis. 3. Future Generalization”, Vestn. St. Petersburg Univ.: Math. 51, issue 2, 164–168 (2018). https://doi.org/10.3103/S1063454118020085
Babenko V.F., “Extreme problems of approximation theory and inequalities for permutations”, Reports of the USSR Academy of Sciences 290(5), 1033–1036 (1986). (In Russian)
Babenko V.F., “Exact inequalities for norms of intermediate derivatives of half-integer order and some of their applications”, Reports of the Ukrainian Academy of Sciences 2, 23–26 (1995). (In Russian)
Babenko V.F., “On inequalities for the norms of intermediate derivatives on a finite interval”, Ukr. Math. J. 47(1), 105–107 (1995). (In Russian)
Babenko V.F., Kofanov V.A., Pichugov S.A., ”On additive inequalities for norms of intermediate derivatives”, RAS reports 356(2), 154–156 (1997). (In Russian)
Burenkov V.I., “On exact constants in inequalities for the norms of intermediate derivatives on a finite interval”, Trudy Math. Instituta im. V. A. Steklova AN SSSR 156, 22–29 (1980). (In Russian)
Kofanov V.A., “On inequalities of Landau — Kolmogorov — Hrmander type on a segment and real line”, Ukr. Math. J. 52(12), 1676–1688 (2000). (In Russian)
Chen W., ”Landau — Kolmogorov inequality on a finite interval”, Bull. Austral. Math. Soc. 48, 485–494 (1993). https://doi.org/10.1017/S000497270001594X
Shadrin A.U., “On exact constants in inequalities between L1-norms of derivatives on a finite interval”, RAS reports 326(1), 150–153 (1992). (In Russian)
Dyn’kin E.M., “Pseudoanalytic extensions of smooth functions. The uniform scale”, Amer. Math. Soc. Transl. 115, 33–38 (1980). https://doi.org/10.1090/trans2/115/02
Dyn’kin E.M., “The pseudoanalytic extensions”, J. Anal. Math. 60, 45–70 (1993). https://doi.org/10.1007/BF03341966
Downloads
Published
2019-11-28
How to Cite
Silvanovich, O. V., & Shirokov, N. A. (2019). Entire functions of order 1/2 in the approximation of functions on the semiaxis. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 6(4), 627–635. https://doi.org/10.21638/11701/spbu01.2019.408
Issue
Section
Mathematics
License
Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
