Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness

We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for r = 2, k = 1 or r = 3, k = 1, 2 and arbitrary q, p [1, ], the following unimprovable inequality holds for functions :

Kolmogorov-type inequalities for mixed derivatives of functions of many variables

Let = (₁,...,_d) be a vector with positive components and let D be the corresponding mixed derivative (of order _j with respect to the jth variable). In the case where d > 1 and 0 < k < r are arbitrary, we prove that

Power Invariants of a Union of Coaxial Prisms

The paper is an addition to the paper of Yu. I. Babenko and V. A. Zalgaller published in the same volume. It gives a condition under which the set of all vertices of several coaxial prisms inscribed in a sphere in has power invariants I₁,...,I_n. A finite set in with 11 invariants is constructed. It is also proved that unions of prisms yield finite sets in with any preassigned number n of invariants with alternating signs. Bibliography: 5 titles.     

Inequalities for upper bounds of functionals on the classes w r H ω and their applications

We show that the well-known results on estimates of upper bounds of functionals on the classes W ^r H ^ω 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 W ^r H ^ω , 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 H ^ω on the derivatives of trigonometric polynomials or polynomial splines in terms of the L ^ϱ -norms of these polynomials and splines.