On Some Properties of Moduli of Smoothness with Jacobi Weights

Abstract

We discuss some properties of the moduli of smoothness with Jacobi weights that we have recently introduced and that are defined asand α, β > −1/p if 0 < p < ∞, and α, β ≥ 0 if p = ∞. We show, among other things, that for all m, n ∈ N, 0 < p ≤ ∞, polynomials P n of degree < n and sufficiently small t, ω ϕ m,0 (P n , t) α,β,p ∼ tω ϕ m−1,1 (P ′ n , t) α,β,p ∼ · · · ∼ t m−1 ω ϕ 1,m−1 (P (m−1)where w α,β (x) = (1 − x) α (1 + x) β is the usual Jacobi weight.In the spirit of Yingkang Hu's work, we apply this to characterize the behavior of the polynomials of best approximation of a function in a Jacobi weighted

Simultaneous approximation by Bernstein polynomials with integer coefficients

“…). Throughout c denotes positive constants, whose value is independent of f and n. Instead of ω 2 ϕ (f, t) we can use the moduli defined and considered in [11,12], [10,15,16,17,18,19,22], or [9].…”