Weighted moduli of smoothness of k-monotone functions and applications

摘要

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.