Best trigonometric approximation in Lp, 1 p ∞, is characterized by a modulus of smoothness, which is equivalent to zero if the function is a trigonometric polynomial of a given degree. The characterization is similar to the one given by the classical modulus of smoothness. The modulus possesses properties similar to those of the classical one.

The uniform weighted approximation errors of Baskakov-type operators are characterized for weights of the form x 1 + x γ 0 (1 + x) γ∞ for γ0, γ∞ ∈ [−1, 0]. Direct and strong converse theorems are proved in terms of the weighted K-functional.

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