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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
Article
  • 15 Downloads

Abstract

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.

Keywords

Differentiable Function Quadratic Approximation Dirichlet Kernel 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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