Weighted Jacobi-type moduli revisited and applied

Let $\alpha,\beta\ge0$ and denote by $w(x):=w_{\alpha,\beta}(x):=(1-x)^\alpha(1+x)^\beta$, the Jacobi weight. Let $L_{p,w}[-1,1]$, $1\le p<\infty$, be the space of functions $f$ such that $fw\in L_p[-1,1]$, with the norm $\|f\|_{p,w}:=\left(\int_{-1}^1|f(x)w(x)|^pdx\right)^{1/p}$, and let $L_{\infty,w}:=\{f\in C(-1,1):\lim_{x\to\pm1}f(x)w(x)=0\}$, equipped with the norm $\|f\|_{\infty,w}:=\max_{x\in(-1,1)}|f(x)w(x)|$. We introduce weighted moduli of smoothness for functions $f\in L_{p,w}[-1,1]\cap C^{r-1}(-1,1)$, $r\ge1$, that have an $(r-1)$st absolutely continuous derivative in $(-1,1)$, and such that $f^{(r)}\in L_{p,\varphi^rw}[-1,1]$, where $\varphi(x)=(1-x^2)^{1/2}$. These moduli are equivalent to certain weighted Ditzian--Totik moduli of smoothness, but our definition is more transparent and simpler. We apply these moduli to obtain Jackson-type estimates on the approximation of functions in $L_{p,w}[-1,1]$, $1\le p\le\infty$, by means of algebraic polynomials. We also have inverse theorems that yield characterization of the behavior of the derivatives of the function by means of its degrees of approximation. In particular when $\alpha=\beta=0$, that is, when $w\equiv1$, we are dealing with the ordinary $L_p[-1,1]$, $1\le p<\infty$ and $C[-1,1]$ and characterize the smoothness of the derivatives of $f$, as measured in weighted moduli of smoothness, by means of its degrees of unweighted approximation.