On Constants in the Jackson Stechkin Theorem in the Case of Approximation by Algebraic Polynomials

Abstract

New estimates are proved for the constants J(k, α) in the classical Jackson–Stechkin inequality E_n−1(f) ≤ J(k, α)ω_k(f,απ/n), α > 0, in the case of approximation of functions f ∈ C[−1, 1] by algebraic polynomials. The main result of the paper implies the following two-sided estimates for the constants: 1/2 ≤ J(2k, α) < 10, n ≥ 2k(2k − 1), α ≥ 2.