New Moduli of Smoothness: Weighted DT Moduli Revisited and Applied

Abstract

We introduce new moduli of smoothness for functions $f\in L_p[-1,1]\cap C^{r-1}(-1,1)$, $1\le p\le \infty$, $r\ge 1$, 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 Ditzian–Totik (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.