Equivalence of Weighted DT-Moduli of (Co)convex Functions

Abstract

The paper present new definitions for weighted DT moduli. Similarly, we a general outcome in an equivalence of moduli of smoothness are obtained. It is known that, any $r\in {\mathbb{N}}_{\circ }$ , $0<p\le \infty$, $1\le \eta \le r$ and $\varphi (x)=\sqrt{1-x^2}$, the inequalities ${\omega }_{i+1,r}^{\varphi }(f^{(r)},\Vert {\theta }_{\mathcal{N}}\Vert {)}_{w_{\alpha ,\beta },p}\sim {\omega }_{i,r+1}^{\varphi }(f^{(r+1)},\Vert {\theta }_{\mathcal{N}}\Vert {)}_{w_{\alpha ,\beta },p}$ and ${\omega }_{i+\eta }^{\varphi }(f,\Vert {\theta }_{\mathcal{N}}\Vert {)}_{\alpha ,\beta ,p}\sim \Vert {\theta }_{\mathcal{N}}{\Vert }^{-\eta }{\omega }_{i,2\eta }^{\varphi }(f^{(2\eta )},\Vert {\theta }_{\mathcal{N}}\Vert {)}_{\alpha +\eta ,\beta +\eta ,p}$ are valid.

16 pages, 1 figure, 3 authors, 21 references. arXiv admin note: text overlap with arXiv:2005.07747