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

  • A. L. Grigoryan

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.
Published
25.12.2004
How to Cite
Grigoryan, A. L. “Estimate of the Remainder of the Best Quadratic Approximation of Differentiable Functions by Polynomials”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, no. 12, Dec. 2004, pp. 1691-8, https://umj.imath.kiev.ua/index.php/umj/article/view/3876.
Section
Short communications