Abstract
We show that the well-known results on estimates of upper bounds of functionals on the classes Wr Hω of periodic functions can be regarded as a special case of Kolmogorov-type inequalities for support functions of convex sets. This enables us to prove numerous new statements concerning the approximation of the classes Wr Hω, establish the equivalence of these statements, and obtain new exact inequalities of the Bernstein-Nikol’skii type that estimate the value of the support function of the class Hω on the derivatives of trigonometric polynomials or polynomial splines in terms of the Lϱ-norms of these polynomials and splines.
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