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

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.