Approximation Growth Of The Dirichlet Series Of Zero Order And Power Series

Based on the classical theories and methods of Dirichlet serise with real expo-nents, in this thesis, we make use of the order and the type of Dirichlet series and Taylor entire functions, to further study the following questions:the slow growth of entire function expressed by power series, the approximation and growth problem of accurate zero (R) order and infinite (R - H) order of Dirichlet series in the whole plane, and the approximation and growth problem of zero order Dirichlet series in the half plane. This thesis is made up of four chapters.In the first chapter, we introduces the research background and current status of Dirichlet series domestic and international, and list the preparation knowledge and related definitions.In the second chapter, first of all, we defined the generalized order and general-ized type of Taylor entire function; secondly, we show some interesting relationship on the maximum modulus, the maximum term and the coefficients of Taylor en-tire function; finally, we study the polynomial approximation of entire function in Banach spaces ((B(p, q, k); f), Hardy spaces, Bargeman spaces), the coefficient char-acterization of generalized type of Taylor entire function of slow growth has been obtained in terms of the approximation errors.In the third chapter, the growth of accurate zero (R) order and infinite (R-H) order of Dirichlet series in the whole planed is studied. The error in approximating Dirichlet series by Dirichlet polynomials was utilized to evaluate the accurate zero (R) order and infinite (R-H) order when the Dirichlet polynomials were used to approach the Dirichlet series, and two necessary and sufficient conditions of Dirichlet series growth are obtained.In the fourth chapter, the growth of zero order Dirichlet series in the right half plane is studied. Using the Dirichlet polynomials to approximating zero order Dirichlet sries which is absolute convergence in the right half plan, some necessary and sufficient conditions about the relationship between the error in approximating Dirichlet sreies with the order and the type function are obtained.