Ph.D. Tezi Görüntüleme | |||||||||||||||||||||
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| Summary: The aim of this thesis is to study the fundamental properties of Fourier series for a continuous linear representation of SO(2) group in a Banach space and to give some applications of those. This thesis consists of two main chapters. In Chapter 1, some definitions and theorems which are crucial for our study are stated. Chapter 2 contains five parts. In the first part, a linear representation of the SO(2) group in a Banach space, bounded linear representation, continuous linear representation are defined and some properties of them are investigated. In the second part, Fourier series of the continuous linear representation are defined and some properties of them are investigated. In the third part, the infinitesimal generator D and its domain H(D) are defined. The quasi-resolvent operator R_λ:H→H is defined for all λ∈C and some properties of R_λ are investigated. In the fourth part, the theorem on integral for the operator D is given. An analog of the resolvent (= quasi-resolvent) operator of D is defined for points of the spectrum of D and its evident form is given. In the fifth part, theorems on the existence of periodic solutions of a linear differential equation of the nth order with constant coefficients in Banach spaces are obtained and in the case of the existence of periodic solutions, evident forms of all periodic solutions are given.
Key Words: Fourier series; Linear representations of groups; Infinitesimal generator; Resolvent operator; Periodic solution; Theorem on integral. | |||||||||||||||||||||