  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.
