Kovalenko O. Interpolation and extremal properties of perfect splines and their applications

The thesis is devoted to investigation of interpolation and extremal properties of perfect splines and their analogues and to applications of these properties to solutions of some approximation theory problems. Existence of interpolation in the mean periodic monosplines and non-symmetric perfect splines and some of their extremal properties were proved. Sharp Kolmogorov type inequalities for classes of functions defined on half-line that are determined by continuous non-increasing positive majorants of functions and their higher derivatives were obtained. New Kolmogorov type comparison theorems for classes of functions that are determined by constraints for the norms of several higher derivatives were obtained. Solutions of optimal recovery problems on the classes of functions that are determined by constraints for the norms of several higher derivatives were obtained. Kolmogorov problem about the existence of a function with given norms of derivatives on the classes of multiple monotone functions and the class of absolute monotone functions for the case of arbitrary norms was solved.