Inequalities between best polynomial approximations and some smoothness characteristics in the space L and widths of classes of functions.

We obtain exact constants in Jackson-type inequalities for smoothness characteristics Λ( f), k ∈ N, defined by averaging the kth-order finite differences of functions f ∈ L. On the basis of this, for differentiable functions in the classes L, r ∈ N, we refine the constants in Jackson-type inequalities containing the kth-order modulus of continuity ω. For classes of functions defined by their smoothness characteristics Λ( f) and majorants Φ satisfying a number of conditions, we calculate the exact values of certain n-widths. [ABSTRACT FROM AUTHOR]