The Growth Of Multiple Laplace-Stieltjes Transform

Laplace-Stieltjes transform is a generalization of Laplace transform and Dirichlet series.The growth of Laplace-Stieltjes transform is not only a key element of it but also the basic of study on the value distribution.In this thesis we study the convergence of Laplace-Stieltjes transform.Basing on it,using the Knopp-Kojima method,we define the maximum modulus,the maximum term and the coefficients,and study the growth,which is a generalization of the corresponding results of 1 dimensional Laplace-Stieltjes transform and 1 dimensional Dirichlet series.The specific contents are as follows.In the first chapter,the research background and situation of multiple Laplace-Stieltjes transform are introduced.Then we give some necessary notations.At last,we give the definition of multiple Laplace-Stieltjes transform.In the second chapter,several kinds of convergence of Laplace-Stieltjes transform are introduced and their properties are studied.Then the associated abscissas of several kinds of convergence are introduced.At last,the estimation of the associated abscissas of several kinds of convergence are studied,basing on which we study the growth of multiple Laplace-Stieltjes transform in the next chapter.In the third chapter,the definition of the maximum modulus,the maximum term and the coefficients of multiple Laplace-Stieltjes transform are studied.Then a special class of functions is introduced,based on which the generalized order is defined to describe the growth of multiple Laplace-Stieltjes transform.At last,the relation between the generalized order and the coefficients is obtained.Some related results of functions of bounded variation in the sense of Vitality and Riemann-Stieltjes integral that we need in this thesis are included in the appendix.