On a problem of optimal recovery and Kolmogorov type inequalities on an interval

Abstract

We consider an optimal recovery problem for the k-th derivative of the function on an interval from the information on the function itself, given in the mean square metric. As a consequence of the solution we prove one Kolmogorov type inequality for derivatives on an interval and demonstrate that the constant in this inequality can be reduced by considering particular subsets of the function class.Optimal recovery problem first appeared in the paper of Smolyak [15] and has been widely developed in [12,[9][10][11]. The problems of this kind are also considered in [3]. Based on the general principles of extremal problems the new approach can be found in [5,13,7,6], as well as some results in this area. In the papers [8,4] authors obtained some inequalities for derivatives and showed that the problem of finding the exact constants in such inequalities can be formulated and efficiently solved as the corresponding optimal recovery problem. In this paper we develop their approach and prove one Kolmogorov type inequality for derivatives (originally obtained in [14] and discussed in the paragraph 5.3 of the book [2]) as a consequence of the solution of the optimal recovery problem. Moreover, we show that the constant in this inequality, which