New moduli of smoothness

TL;DR

This paper introduces and analyzes a new modulus of smoothness involving weighted differences, providing insights into function smoothness and approximation properties with various weights.

Contribution

It proposes a novel modulus of smoothness incorporating weights and explores its properties and generalizations, advancing the theoretical understanding of function approximation.

Findings

Properties of the new modulus are established.

Connections with existing moduli are analyzed.

Generalizations with different weights are considered.

Abstract

In this paper, we discuss various properties of the new modulus of smoothness \[ \omega^\varphi_{k,r}(f^{(r)},t)_p := \sup_{0 < h\leq t}\|\mathcal W^r_{kh}(\cdot) \Delta_{h\varphi(\cdot)}^k (f^{(r)},\cdot)\|_{L_p[-1,1]}, \] where $\mathcal W_\delta(x) = \bigl((1-x-\delta\varphi(x)/2) (1+x-\delta\varphi(x)/2)\bigr)^{1/2}. $ Related moduli with more general weights are also considered.