Dmytryshyn M. Approximation spaces associated with entire exponential type vectors

In the thesis the new classes of approximation spaces associated with unbounded operators in the normed spaces are introduced and described, in the context of spectral approximations and characterization of different classes of functions in terms of their best approximations by entire exponential type functions in the functional spaces. The inequalities of Bernstein and Jackson type in terms of quasi-norms of Besov type approximation spaces with exact values of constants are proved, which give analytical estimates of the best approximations by entire exponential type functions of unbounded operator, in particular, spectral approximations in the case of an operator with the point spectrum. The interpolation spaces of entire exponential type vectors of a closed operator generated by real and complex interpolation methods are described, and their properties are established. The approximation spaces associated with regular elliptic differential operators, degenerate elliptic differential operators, and generalized Legendre differential operators are determined, and the corresponding Bernstein and Jackson type inequalities characterizing the approximation by entire exponential type functions and root functions in Lebesgue spaces are established. The tensor products of approximation spaces associated with sets of closed operators, in particular, positive operators, regularly elliptic differential operators and generalized Legendre differential operators are determined, and their interpolation properties are established. The inequalities of Bernstein and Jackson type on tensor products of approximation spaces of Besov type are proved with exact estimates of the best approximations by the root functions of operators in different functional spaces.