Fine-grained evaluations of the best estimates for smooth functions in $C_{2\pi}$ in terms of linear combinations of modules of continuity of their derivatives

Abstract

UDC 517.5

For the best approximations of $e_{n-1}(f)$ functions in $C^1_{2\pi}$ by trigonometric polynomials, Zhuk proved the exact Jackson inequality $e_{n-1}(f)\leqslant \dfrac{\pi}{4n}\omega\left(f',\dfrac{\pi}{n}\right)$.
In this paper, we prove the following version of Jackson's exact inequality: $e_{n-1}(f)\leqslant \dfrac{\pi}{4n}\left(\dfrac{1}{2}\omega\left(f',\dfrac{\pi}{2n}\right)+\dfrac{1}{2}\omega\left(f',\dfrac{\pi}{n}\right)\right)$.