2017
Том 69
№ 6

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

Grigoryan A. L.

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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.

English version (Springer): Ukrainian Mathematical Journal 56 (2004), no. 12, pp 1998-2006.

Citation Example: Grigoryan A. L. Estimate of the Remainder of the Best Quadratic Approximation of Differentiable Functions by Polynomials // Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1691-1698.

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