Kovalenko O. Inequalities for derivatives and extremal problems of Approximation Theory in metric spaces

2. The dissertation is devoted to classical problems of Approximation Theory, in particular to sharp Landau-Kolmogorov type inequalities, to the Stechkin problem about approximation of unbounded operators by bounded ones, to the problem to find the modulus of continuity of operators, as well as to problems of optimal recovery of operators and functionals given exact or inexact information, in particular to problems of optimization of cubature formulae. Chapter 1 is devoted to a study of extremal problems in spaces of functions with values in L-spaces i.e., in semilinear metric spaces with additional axioms that connect the metric with the algebraic operations. Such approach allows to include into consideration various classes of functions, in particular classes of multi-valued and fuzzy-valued functions, as well as classes of functions with values in normed spaces, including classes of random processes. We obtain a generalization of the Korneichuk-Stechkin lemma for functions with values in L-spaces. We prove sharp Ostrowski-type inequalities and solve problems of optimal recovery of operators and functionals on various classes of L-space valued functions. Chapter 2 is devoted to extremal problems for operators that act on the Sobolev classes of multivariate functions. For these classes of functions we consider the problem of the best approximation of the hypersingular integral operator D using bounded operators. We also prove sharp Landau-type inequalities in the additive form that estimate the uniform norm of Df via the uniform norm of the function f and an integral norm of its gradient. We also compute the modulus of continuity of the operator D and solve the problem of optimal recovery of this operator given the values of its arguments known with an error. We solve the problem of optimal recovery of the integral with unit and non-unit weight on different domains of definitions of the functions. Chapter 3 is devoted to inequalities that estimate the deviation between the value of a function at some point and the mean value of the function, via some characteristics of the function. Such inequalities are often called Ostrowski type inequalities. Inequalities of this kind can be applied to solutions of other extremal problems of Approximation Theory, in particular for classes of functions of low smoothness they can be applied to problems of optimal recovery and to prove Landau--Kolmogorov type inequalities. For multivariate functions we propose a new definition of the notion of bounded variation and prove sharp Ostrowski-type inequalities. For a class of random processes that is determined by a majorant of modulus of continuity of the processes, we prove a sharp Ostrowski type inequality that estimates the deviation between the integral of the process and the value of the process at a random. Using this inequality, we solve a problem of optimal recovery of the integral of the random process, given the values of the process at n random moments of time. Chapter 4 is devoted to the inequalities for derivatives of Landau-Kolmogorov-type, of Nagy type, and related extremal problems. We obtain sharp Nagy type inequalities that estimate the uniform norm of a function from a Sobolev space using the L_p-norm of its gradient and some seminorm that is defined on the space of locally integrable on an open cone functions. We find the modulus of continuity of a higher order differentiation operator on the classes of functions defined on a half-line that are determined by (non-constant) majorants of the functions and their higher derivatives. We prove a snake theorem that guarantees existence of perfect spline analogues that oscillate maximally. These splines are extremal in the problem to find the modulus of continuity of the differentiation operator.