Best approximations and moduli of smoothness of functions and their derivatives in $L_p$, $0<p<1$

Abstract: Several new inequalities for moduli of smoothness and errors of the best approximation of a function and its derivatives in the spaces $L_p$, $0<p<1$, are obtained. For example, it is shown that for any $0<p<1$ and $k,\,r\in \mathbb{N}$ one has $ \omega_{r+k}(f,\d)p\leq C({p,k,r})\d^{r+\frac{1}{p}-1}(\int_0^\d\frac{\omega{k}(f^{(r)},t)_p^p}{t^{2-p}}{\rm d}t)^\frac{1}{p},$ where the function $f$ is such that $f^{(r-1)}$ is absolutely continuous. Similar inequalities are obtained for the Ditzian-Totik moduli of smoothness and the error of the best approximation of functions by trigonometric and algebraic polynomials and splines. As an application, positive results about simultaneous approximation of a function and its derivatives by the mentioned approximation methods in the spaces $L_p$, $0<p<1$, are derived.