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

Abstract

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 $E(X,{\Pi }_n{)}_{w,q}:={\sup }_{f\in X}{\inf }_{P\in {\Pi }_n}\Vert w(f-P){\Vert }_{L_q}$, 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 $\Vert w_{\alpha ,\beta }f{\Vert }_{L_p}=1$. In this paper, we determine the exact behavior (in terms of $\delta$) of ${\sup }_{f\in S_p^{\alpha ,\beta }\cap M^k}{\omega }_{\phi }^k(f,\delta {)}_{w_{\alpha ,\beta },L_q}$ for $1\le p,q\le \infty$ (the interesting case being $q<p$ as expected) and $\alpha ,\beta >-1/p$ (if $p<\infty$) or $\alpha ,\beta \ge 0$ (if $p=\infty$). It is interesting to note that, in one case, the behavior is different for $\alpha =\beta =0$ and for $(\alpha ,\beta )\ne (0,0)$. 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 $\alpha ,\beta \ge 0$.