In this paper, we discuss various properties of the new modulus of smoothness [ omega^varphi_{k,r}(f^{(r)},t)_p := sup_{0 < hleq t}|mathcal W^r_{kh}(cdot) Delta_{hvarphi(cdot)}^k (f^{(r)},cdot)|_{L_p[-1,1]}, ] where $mathcal W_delta(x) = bigl((1-x-de
ltavarphi(x)/2) (1+x-deltavarphi(x)/2)bigr)^{1/2}. $ Related moduli with more general weights are also considered.
We introduce new moduli of smoothness for functions $fin L_p[-1,1]cap C^{r-1}(-1,1)$, $1le pleinfty$, $rge1$, 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 $varph
i(x)=(1-x^2)^{1/2}$. 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.
