Weighted moduli of smoothness of k-monotone functions and applications

Let omega(k)(phi)(f, delta)omega,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 kth divided differences are nonnegative for all selections of k +1 distinct points in (-1, 1), and denote epsilon(X, P-n)w,q :=suP(f is an element of X) inf P is an element of P-n parallel to w(f - P)parallel to L-q, where P-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 parallel to W-alpha,W-beta f parallel to L-p = 1. In this paper, we determine the exact behavior (in terms of delta) of supf is an element of S-p(alpha,beta)boolean AND M-k omega(k)(phi)(f, delta)(omega alpha,beta),Lq for fEs7,43n.mk 1 <= p, q <= infinity (the interesting case being q < p as expected) and alpha, beta > -1/p (if p < infinity) or alpha,beta >= 0 (if p = no). It is interesting to note that, in one case, the behavior is different for alpha = beta = 0 and for (alpha, beta) not equal (0, 0). Several applications are given. For example, we determine the exact (in some sense) behavior of epsilon(Mk boolean AND S-p(alpha,beta), P-n)wa,fl,Lq for alpha,beta >= 0. (C) 2014 Elsevier Inc. All rights reserved.