The thesis focuses on the solution of Lagrange, Hermite, and Hermite-Birkhoff operator interpolation problems in Hilbert and finite-dimensional Euclidean spaces, and aims to study the interpolation convergence, obtain the accuracy estimation of interpolation formulas when initial information is disturbed, and outline the conditions for the solvability of the interpolation problem in case of under determinacy.
The thesis includes an introduction, five chapters, a list of references, and appendices.
The first chapter reveals the statements of operator interpolation problems and the analysis of the relevant scholarly papers.
The second chapter shows that in general, the interpolation does not converge to the polynomial operator for the Lagrange interpolation problem in the Hilbert space, however, the linear interpolation is convergent. It is proved that in a separable Hilbert space with a Gaussian measure, the interpolation error can be made arbitrarily small. The research ascertains that the interpolation stays convergent when there is a certain choice of interpolation nodes. The paper investigates Lagrange interpolation formulas for a nonlinear operator in a Hilbert space with a measure and shows that interpolants asymptotically retain polynomials of the corresponding degree. These formulas are much simpler in design. The availability of estimation accuracy enables such interpolants to be used for approximating polynomial and integer operators. The study reveals the interpolation accuracy estimation of polynomial and integer operators when initial information is disturbed. Also, the number of interpolation nodes is obtained and shown that its exceeding does not improve the accuracy of interpolation formulas. Similar theorems are proved for polynomial and integer functionals that are defined on the spaces L_2(0,1) and W_1^2(0,π) when there is an approximate calculation of scalar products in interpolation formulas. The investigation assumes the conditions for the invariant solvability of the Lagrange interpolation problem in Euclidean space in case of under-determinacy. Accordingly, it is stated that the solution is the only one and has the minimum norm among all interpolants. The study investigates the accuracy of the Lagrange formula on polynomials of the appropriate degree in a linear infinite-dimensional space with a scalar product and in a finite-dimensional Euclidean space, and it shows that this interpolation formula contains fundamental Lagrange polynomials.
In the third chapter, in linear topological space, an interpolation Newton-type polynomial of integral form is considered. The chapter proves a theorem concerning the conditions of continual nodes' existence for the chosen interpolant. The obtained result for operators of many variables is generalized.
The fourth chapter reveals the Hermite interpolation polynomial in Hilbert space with a measure when the operator values are given and its first Gateaux differentials at nodes are chosen in a certain way. It is proved that the interpolant has a minimum norm on the set of Hermite interpolants. The theorem on the minimum norm interpolant is generalized for the case when there are Gateaux differentials up to a certain order in the interpolation nodes. Furthermore, the chapter considers the issue of interpolation accuracy and convergence to the polynomial operator. It is shown that Hermite and Hermite-Birkhoff interpolants asymptotically preserve polynomials of the corresponding degree. These formulas are much simpler in design and the availability of estimation accuracy enables such interpolants to be used for approximating polynomial, integer, and differentiable operators. The investigation outlines the Hermite interpolation problems when it is given the value of the function of many variables and the value of its corresponding first- and second-order Gateaux differentials in the interpolation nodes. It is also revealed that these problems have a unique minimum norm solution. The study demonstrates the conditions of invariant solvability and problem solutions in case of under-determinacy. In the normalized infinite-dimensional linear and finite-dimensional Euclidean spaces, it is shown that the Hermite interpolation polynomial of minimum norm contains fundamental polynomials. The accuracy of Hermite interpolation formulas on polynomials of the corresponding degree is studied. The theorem on the Hermite-Birkhoff interpolant of the minimal norm in a Hilbert space with the measure is proved.
The fifth chapter ascertains the new compatibility criteria of the linear system of equations and inequalities, which are equivalent to Kronecker-Capelli's and Chernikov's theorems correspondingly, and are related to the conditions of existence of a linear interpolation polynomial in Euclidean spaces. The conditions for the existence of the nonlinear (polynomial) equation solution are found. Examples of solving problems using the obtained results are considered.
