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

Let $\omega_\varphi^k(f,\delta)_{w,L_q}$ 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 $\| w_{\alpha,\beta}f\|_{L_p}=1$. 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 $1\leq p, q\leq \infty$ (the interesting case being $q<p$ as expected) 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 $(\alpha,\beta)\neq (0,0)$. 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 $\alpha,\beta \geq 0$.