Equivalence of Weighted DT-Moduli of (Co)convex Functions (2006.05258v1)

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 $r \in \mathbb{N}{\circ}$ , $0<p \leq \infty$, $1 \leq \eta \leq r$ and $\phi(x)=\sqrt{1-x^2}$, the inequalities $\omega^{\phi}{i+1,r} \; (f^{(r)}, | \theta_{\mathcal{N}} |){w{\alpha, \beta}, p} \sim \omega^{\phi}_{i,r+1} \; (f^{(r+1)}, | \theta_{\mathcal{N}} |){w{\alpha, \beta}, p}$ and $\omega^{\phi}_{i+\eta} \; (f, | \theta_{\mathcal{N}} |){\alpha, \beta, p} \sim | \theta{\mathcal{N}} |^{- \eta} \omega^{\phi}_{i, 2 \eta} \; (f^{(2 \eta)}, | \theta_{\mathcal{N}} |)_{\alpha+ \eta, \beta+ \eta, p}$ are valid.