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

TL;DR

This paper introduces new weighted DT moduli definitions and establishes their equivalence, providing a unified framework for measuring smoothness of (co)convex functions with potential applications in approximation theory.

Contribution

It presents novel definitions for weighted DT moduli and proves their equivalence, extending existing inequalities for smoothness measures of (co)convex functions.

Findings

Established equivalence of different weighted DT moduli.

Extended inequalities relating moduli of different orders.

Provided a general framework for smoothness measurement.

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.