Abstract
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 ≤ ∞.
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