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

  • Yu. P. Babich Ukrainian state university of science and technologies, Dnipro
  • T. F. Mikhailova Ukrainian state university of science and technologies, Dnipro
Keywords: modules of the non-procurrency, trigonometric polynomial

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: $e_{n-1}(f)\leqslant \dfrac{\pi}{4n}\left(\dfrac{1}{2}\omega\left(f',\dfrac{\pi}{2n}\right)+\dfrac{1}{2}\omega\left(f',\dfrac{\pi}{n}\right)\right)$.

References

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Published
23.05.2022
How to Cite
BabichY. P., and MikhailovaT. F. “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”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 4, May 2022, pp. 569 -72, doi:10.37863/umzh.v74i4.7124.
Section
Short communications