Ukrainian Mathematical Journal

, Volume 56, Issue 12, pp 1998–2006 | Cite as

Estimate of the Remainder of the Best Quadratic Approximation of Differentiable Functions by Polynomials

  • A. L. Grigoryan


We establish lower and upper bounds for the quantity
$$C_m^q (W^r ,x) = \mathop {\sup }\limits_{f \in W^r } \left| {f(x) - T_m (x,f)} \right|,$$
, where
$$T_m (x,f) = \frac{2}{q}\mathop \sum \limits_{l = 0}^{q - 1} \;f(x_l )D_m (x - x_l ),\quad q \in \mathbb{N},\quad q > 2m,\quad x_l = \frac{{2\pi l}}{q},\quad l = 0,\;1,\;...\;,\;q - 1,$$
, and D m (t) is the Dirichlet kernel, for the class W r of 2π-periodic functions, whose rth derivative satisfies the condition |f r (x)| ≤ 1.


Differentiable Function Quadratic Approximation Dirichlet Kernel 
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  1. 1.
    M. L. Kalashnikov, “On polynomials of the best quadratic approximation in a given system of points,” Dokl. Akad. Nauk SSSR, 105, No.4, 634–636 (1955).Google Scholar
  2. 2.
    G. P. Gubanov, “Approximation of functions by trigonometric polynomials of the best quadratic approximation,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 12, 22–29 (1970).Google Scholar
  3. 3.
    S. M. Nikol’skii, “Interpolation trigonometric polynomials with equidistant points of interpolation,” Dokl. Akad. Nauk SSSR, 31, No.3 (1941).Google Scholar
  4. 4.
    N. P. Korneichuk, Extremal Problems in Approximation Theory [in Russian], Nauka, Moscow (1976).Google Scholar
  5. 5.
    Yu. N. Subbotin, “On relationship between finite differences and corresponding derivatives,” Tr. Mat. Inst. Akad. Nauk SSSR, 78, 24–42 (1965).Google Scholar
  6. 6.
    A. L. Grigoryan, “Discrete Lebesgue constants,” Mat. Zametki, 34, No.6, 857–866 (1983).Google Scholar
  7. 7.
    A. L. Grigoryan, “Best quadratic approximation of differentiable functions by polynomials,” Izv. Nats. Akad. Nauk Armenii, 35, No.5, 25–33 (2000).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • A. L. Grigoryan
    • 1
  1. 1.Erevan BranchTernopol Academy of National EconomyErevanArmenia

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