Parfinovych N. Splines in extremal problems of Approximation Theory, inequalities for derivatives and their applications

In the Thesis we find exact values of the best approximations of classes of differentiable periodic functions by subspaces of splines with defect 2 having uniformly distributed nodes and by subspaces of splines with minimal defect having non-uniformly distributed nodes. We show that these subspaces are extremal for Kolmogorov width of specified classes. We find exact values of the best approximations of convolution classes of periodic functions from an arbitrary rearrangement-invariant set with variation diminishing kernels by subspaces of generalized splines. We show that subspaces of generalized splines are extremal for Kolmogorov width of specified convolution classes. We establish the series of sharp Kolmogorov type inequalities for univariate and multivariate functions. We obtain the solution of the Stechkin problem of the best approximation of unbounded fractional differentiation operators by bounded ones and the problem of the best recovery of fractional differentiation operator on classes of function, given with an error.