Function Spaces Represented By Entire Dirichlet Series And The Growth Of Generalized Laplace-Stieltjes Transforms

In this thesis,we investigate function spaces represented by entire Dirichlet series and the growth of generalized Laplace—Stieltjes transforms.This thesis is divided into four chapters.In the first.chapter,we introduce the order and associated knowledge of Dirichlet series and the Laplace-Stieltjes transforms.In the second chapter,the definition of the linear space F represented by entire Dirichlet series is given firstly.Then by using of theories of Dirichlet series and functional analysis,on the condition of norm 1,we prove that F is a commutative and not a division Banach algebra with identity,necessary and sufficient conditions for the element invertible and topological zero divisor are given;on the condition of norm 2,a necessary and sufficient conditions for the continuous linear functional on the space F is given.In chapter 3,we mainly study the growth of entire functions represented by general-ized Laplace—Stieltjes transforms defined in a piecewise smooth Jordan curve starting from origin in half right plane.We define the maximum modulus M(r,F)and the maxi-mum term m(r,F)of F(s)in the circle |z| = r.Then under more general conditions,we study the relation between the coefficient of entire functions F(s)represented by general-ized Laplace—Stieltjes transforms and the growth in the whole plane.In chapter 4,based on chapter 3,we introduce a order of entire functions F(s)represented by generalized Laplace-Stieltjes transforms,then study the relationship between a order and coefficients.To further study the growth of F(s),we introduce the definition of generalized type and type function,then study the relationship between generalized type and type function and coefficients.