On a problem of optimal recovery and Kolmogorov type inequalities on an interval (Q476138)

The author uses the solution of certain extremal problems in order to solve 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. First, he shows an explicit expression for the error of the optimal recovery and for optimal methods in terms of the Lagrange multipliers and an orthonormal basis of Jacobi polynomials, respectively (see Theorem 1). Secondly, the author proceeds to the application of Theorem 1 to corresponding inequalities for derivatives (see Proposition 1). Thirdly, the author obtains an analogous of Theorem 1 for the case in which the function belongs to the Sobolev class \(W_0^r\) (cf. Theorem 2), and finally as a consequence of Theorem 2, the author proves a Kolmogorov type inequality for derivatives on an interval and shows that the constant in this inequality can be reduced by considering particular subsets of a weighted Sobolev class.