Weighted moduli of smoothness of k-monotone functions and applications

Let ωk φ( f, δ)w,Lq 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 kth divided differences are nonnegative for all selections of k + 1 distinct points in (−1, 1), and denote E(X, Pn)w,q := sup f ∈X infP∈Pn ∥w( f − P)∥Lq , where Pn is the set of algebraic polynomials of degree at most n. Additionally, let wα,β (x) := (1 + x)α(1 − x)β be the classical Jacobi weight, and denote by S p the class of all functions such that wα,β f Lp = 1. In this paper, we determine the exact behavior (in terms of δ) of sup f ∈S p ∩Mk ωk φ( f, δ)wα,β ,Lq for 1 ≤ p, q ≤ ∞ (the interesting case being q < p as expected) and α, β > −1/p (if p < ∞) or α, β ≥ 0 (if p = ∞). It is interesting to note that, in one case, the behavior is different for α = β = 0 and for (α, β) ≠ (0, 0). Several applications are given. For example, we determine the exact (in some sense) behavior of E(Mk ∩ S p , Pn)wα,β ,Lq for α, β ≥ 0. c ⃝ 2014 Elsevier Inc. All rights reserved. MSC: 41Axx