New moduli of smoothness: weighted DT moduli revisited and applied

TL;DR

This paper introduces new, simpler moduli of smoothness for functions on [-1,1], establishing their equivalence to weighted DT moduli and applying them to derive Jackson-type approximation estimates and inverse theorems for polynomial approximation.

Contribution

The paper presents a more transparent and simpler definition of moduli of smoothness, linking them to polynomial approximation without weights and providing constructive smoothness characterizations.

Findings

New moduli are equivalent to weighted DT moduli.

Derived Jackson-type approximation estimates.

Proved inverse theorems for smoothness characterization.

Abstract

We introduce new moduli of smoothness for functions $f\in L_p[-1,1]\cap C^{r-1}(-1,1)$, $1\le p\le\infty$, $r\ge1$, that have an $(r-1)$st locally absolutely continuous derivative in $(-1,1)$, and such that $\varphi^rf^{(r)}$ is in $L_p[-1,1]$, where $\varphi(x)=(1-x^2)^{1/2}$. These moduli are equivalent to certain weighted DT moduli, but our definition is more transparent and simpler. In addition, instead of applying these weighted moduli to weighted approximation, which was the purpose of the original DT moduli, we apply these moduli to obtain Jackson-type estimates on the approximation of functions in $L_p[-1,1]$ (no weight), by means of algebraic polynomials. Moreover, we also prove matching inverse theorems thus obtaining constructive characterization of various smoothness classes of functions via the degree of their approximation by algebraic polynomials.