A Characterization of Smoothness in Terms of Approximation by Algebraic Polynomials in LP Original Research Article

We prove direct and inverse theorems for the classical modulus of smoothness and approximation by algebraic polynomials in Lp[−1, 1]. These theorems contain the well-known theorems of A. Timan, V. Dzyadyk, G. Freud, and Yu. Brudnyi as special cases when p = ∞. They also provide a characterization of the spaces Lip(α, p) (Lipschitz spaces in Lp) for 0 < α < ∞, 1 ≤ p ≤ ∞.