Abstract
It is proved that, in the space C2π, for all k, n ∈ ℕ,n > 1, the following inequalities hold:
where e n−1(f) is the value of the best approximation of f by trigonometric polynomials and ω 2(f, h) is the modulus of smoothness of f. A similar result is also obtained for approximation by continuous polygonal lines with equidistant nodes.
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N. P. Korneichuk, “The exact constant in Jackson’s theorem on best uniform approximation of continuous periodic functions,” Dokl.Akad. Nauk SSSR 145(3), 514–515 (1962) [Soviet Math. Dokl. 3 (3), 1040–1041 (1962)].
N. P. Korneichuk, “On the sharp constant in Jackson’s inequality for continuous periodic functions,” Mat. Zametki 32(5), 669–674 (1982) [Math. Notes 32 (5), 818–821 (1983)].
N. I. Chernykh, “On Jackson’s inequality in L 2,” in Trudy Mat. Inst. Steklov, Vol. 88: Approximation of Functions in the Mean, Collection of papers (Nauka, Moscow, 1967), pp. 71–74 [Proc. Steklov Inst. Math. 88, 75–78 (1967)].
N. I. Chernykh, “Best approximation of periodic functions by trigonometric polynomials in L 2,” Mat. Zametki 2(5), 513–522 (1967) [Math. Notes 2 (5), 803–808 (1968)].
N. I. Chernykh, “Jackson’s inequality in L p(0, 2π), (1 ≤ p < 2), with sharp constant,” in Trudy Mat. Inst. Steklov, Vol. 198: Proceedings of an All-Union School on the Theory of Functions, Miass, July 1989 (Nauka, Moscow, 1992), pp. 232–241 [Proc. Steklov Inst. Math. 198, 223–231 (1994)].
V. V. Shalaev “On the approximation of continuous periodic functions by trigonometric polynomials,” in Studies of Problems of Current Interest Dealing with Summation and Approximations of Functions and Their Applications (Dnepropetrovsk. Univ., Dnepropetrovsk, 1979), pp. 39–43 [in Russian].
N. P. Korneichuk, Exact Constants in Approximation Theory (Nauka, Moscow, 1987) [in Russian].
A. A. Ligun, “Sharp constants in inequalities of Jackson type,” in Special Questions of Approximation Theory and Optimal Control of Distributed Systems (Vishcha Shkola, Kiev, 1990), pp. 3–74 [in Russian].
V. V. Zhuk, “Some sharp inequalities between best approximations and moduli of continuity,” Vestnik Leningrad. Univ. Mat. Mekh. Astronom., No. 1, 21–26 (1974).
V. V. Zhuk, Approximation of Periodic Functions (Izd. Leningradsk. Univ., Leningrad, 1982) [in Russian].
A. G. Babenko and Yu. V. Kryakin, “Integral approximation of the characteristic function of the interval and Jackson’s inequality in C(\( \mathbb{T} \)),” Trudy Inst. Mat. Mekh. (Ural Branch, Russian Academy of Sciences) (2009), Vol. 15, pp. 59–65 [in Russian].
N. P. Korneichuk, Splines in Approximation Theory (Nauka, Moscow, 1984) [in Russian].
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Original Russian Text © S. A. Pichugov, 2013, published in Matematicheskie Zametki, 2013, Vol. 93, No. 6, pp. 932–938.
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Pichugov, S.A. Sharp constant in Jackson’s inequality with modulus of smoothness for uniform approximations of periodic functions. Math Notes 93, 917–922 (2013). https://doi.org/10.1134/S0001434613050295
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DOI: https://doi.org/10.1134/S0001434613050295