Equivalence of Weighted DT-Moduli of (Co)convex Functions
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 $r \in \mathbb{N}_{\circ}$ , $0<p \leq \infty$, $1 \leq η\leq r$ and $ϕ(x)=\sqrt{1-x^2}$, the inequalities $ω^ϕ_{i+1,r} \; (f^{(r)}, \| θ_{\mathcal{N}} \|)_{w_{α, β}, p} \sim ω^ϕ_{i,r+1} \; (f^{(r+1)}, \| θ_{\mathcal{N}} \|)_{w_{α, β}, p}$ and $ω^ϕ_{i+η} \; (f, \| θ_{\mathcal{N}} \|)_{α, β, p} \sim \| θ_{\mathcal{N}} \|^{- η} ω^ϕ_{i, 2 η} \; (f^{(2 η)}, \| θ_{\mathcal{N}} \|)_{α+ η, β+ η, p}$ are valid.
math.FA↗