Respondent

Dmytryshyn Marian Ivanovych

Theme

Approximation spaces associated with entire exponential type vectors

Defence Date

14.07.2020

Annotation

The dissertation is devoted to the development of the theory of approximation spaces associated with entire exponential type vectors of unbounded operators in the context of spectral approximations and characterization of different classes of functions in terms of their best approximations by entire exponential type functions.
We introduce the new classes of Besov-type approximation spaces associated with unbounded operators in the Banach space. We show that such spaces are interpolation spaces and we prove the Bernstein and Jackson inequalities in terms of quasi-norms of approximation spaces. The explicit dependence of the constants on the parameters of the Besov type space is obtained.
For the differentiation operator in the space Lq(R), we show the application of abstract results in the approximation theory of functions. We prove that the approximation space associated with the differentiation operator coincides with the classical Besov space. As a consequence, classical inequalities of Bernstein and Jackson with exact values of constants are obtained.
The interpolation theory of spaces of entire exponential type vectors associated with unbounded operators in Banach spaces is developed in the context of its applications in the approximation theory of functions, in particular, spectral approximations in the function spaces. The basic properties of interpolation spaces of entire exponential type vectors of closed operator generated by real and complex interpolation methods are defined and established. An interpolation theorem for the spaces of exponential type vectors of unbounded operators is proved on domains of their integer degrees generated by the complex method interpolation.

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