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

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. © 2014 Published by Elsevier Inc.