Multiple modules of continuity and the best approximations of periodic functions in metric spaces

  • S. A. Pichugov Днепропетр. нац. ун-т ж.-д. трансп.

Abstract

It is proved that, under the condition $M_{\Psi} \Bigl( \frac 12\Bigr) < 1$, where $M_{\Psi}$ is a stretching function $\Psi$ in the space $L_{\Psi}$ , the Jackson inequalities $$\sup_n \sup_{f\in L_{\Psi}, f\not = \text{const}} \frac{E_{n-1}(f)_{\Psi} }{\omega_k \Bigl(f, \frac{\pi}n \Bigr)_{\Psi}} < \infty,$$ are true; here, $E_{n-1}(f)_{\Psi}$ is the best approximation of $f$ by trigonometric polynomials of degree at most $n - 1$ and $\omega_k \Bigl(f, \frac{\pi}n \Bigr)_{\Psi}$ is the modulus of continuity of $f$ of order $k$, $k \in N$. We study necessary and sufficient conditions for the function $f$ under which the following relation is true: $E_{n-1}(f)_{\Psi} \asymp \omega_k \Bigl(f, \frac{\pi}n \Bigr)_{\Psi}.$
Published
25.05.2018
How to Cite
PichugovS. A. “Multiple Modules of Continuity and the Best Approximations of Periodic functions in Metric Spaces”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, no. 5, May 2018, pp. 699-07, http://umj.imath.kiev.ua/index.php/umj/article/view/1588.
Section
Research articles