Weighted moduli of smoothness of $k$-monotone functions and applications

Let ${\omega }_{\phi }^k(f,\delta {)}_{w,L_q}$ be the Ditzian-Totik modulus with weight $w$, $M^k$ be the cone of $k$-monotone functions on $(-1,1)$, i.e., those functions whose $k$th divided differences are nonnegative for all selections of $k+1$ distinct points in $(-1,1)$, and denote $E (X, \Pi_n){w,q} := \sup{f\in X} \inf_{P\in\Pi_n}|w(f-P)|{L_q}$,where$\Pi_n$ is the set of algebraic polynomials of degree at most $n$. Additionally, let $w_{\alpha ,\beta }(x):=(1+x{)}^{\alpha }(1-x{)}^{\beta }$ be the classical Jacobi weight, and denote by $S_p^{\alpha ,\beta }$ the class of all functions such that $\mathrm{\parallel }{w}_{\alpha ,\beta }f{\mathrm{\parallel }}_{L_p}=1$. In this paper, we determine the exact behavior (in terms of $\delta$) of $\sup{f\in S_p^{\alpha,\beta}\cap M^k} \omega_\varphi^k(f,\delta){w{\alpha,\beta},L_q}$for$1\leq p, q\leq \infty$(theinterestingcasebeing$q<p$asexpected)and$\alpha,\beta >-1/p$(if$p<\infty$)or$\alpha,\beta\geq 0$(if$p=\infty$). It is interesting to note that, in one case, the behavior is different for $\alpha =\beta =0$ and for $(\alpha ,\beta )\mathrm{\ne }(0,0)$. Several applications are given. For example, we determine the exact (in some sense) behavior of $E (M^k\cap S_p^{\alpha,\beta}, \Pi_n){w{\alpha,\beta},L_q}$for$\alpha,\beta \geq 0$.