New moduli of smoothness: weighted DT moduli revisited and applied

摘要

We introduce new moduli of smoothness for functions $f\in L_p[-1,1]\capC^{r-1}(-1,1)$, $1\le p\le\infty$, $r\ge1$, that have an $(r-1)$st locallyabsolutely 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 areequivalent to certain weighted DT moduli, but our definition is moretransparent and simpler. In addition, instead of applying these weighted modulito weighted approximation, which was the purpose of the original DT moduli, weapply these moduli to obtain Jackson-type estimates on the approximation offunctions in $L_p[-1,1]$ (no weight), by means of algebraic polynomials.Moreover, we also prove matching inverse theorems thus obtaining constructivecharacterization of various smoothness classes of functions via the degree oftheir approximation by algebraic polynomials.