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

Abstract

Let $ω_φ^k(f,δ)_{w,L_q}$ 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 $E (X, Π_n)_{w,q} := \sup_{f\in X} \inf_{P\inΠ_n}\|w(f-P)\|_{L_q}$, where $Π_n$ is the set of algebraic polynomials of degree at most $n$. Additionally, let $w_{α,β}(x) := (1+x)^α(1-x)^β$ be the classical Jacobi weight, and denote by $S_p^{α,β}$ the class of all functions such that $\| w_{α,β}f\|_{L_p}=1$. In this paper, we determine the exact behavior (in terms of $δ$) of $\sup_{f\in S_p^{α,β}\cap M^k} ω_φ^k(f,δ)_{w_{α,β},L_q}$ for $1\leq p, q\leq \infty$ (the interesting case being $q -1/p$ (if $p<\infty$) or $α,β\geq 0$ (if $p=\infty$). It is interesting to note that, in one case, the behavior is different for $α=β=0$ and for $(α,β)\neq (0,0)$. Several applications are given. For example, we determine the exact (in some sense) behavior of $E (M^k\cap S_p^{α,β}, Π_n)_{w_{α,β},L_q}$ for $α,β\geq 0$.