The approximation of Laplace-Stieltjes transforms with finite order

In this paper, we study the irregular growth of an entire function defined by the Laplace-Stieltjes transform of finite order convergent in the whole complex plane and obtain some results about λ-lower type. In addition, we also investigate the problem on the error in approximating entire functions defined by the Laplace-Stieltjes transforms. Some results about the irregular growth, the error, and the coefficients of Laplace-Stieltjes transforms are obtained; they are generalization and improvement of the previous conclusions given by Luo and Kong, Singhal and Srivastava.


Introduction
Dirichlet series f (s) = ∞ n= a n e λ n s , s = σ + it, where  ≤ λ  < λ  < · · · < λ n < · · · , λ n → ∞ as n → ∞; (  ) s = σ + it (σ , t are real variables), a n are nonzero complex numbers. When a n , λ n , n satisfy some conditions, the series () is convergent in the whole plane or the half-plane, that is, f (s) is an analytic function or entire function in the whole plane or the half-plane. In the past few decades, many mathematicians studied the growth and value distribution of the analytic (entire) function defined by Dirichlet series and obtained lots of interesting results (see [-]). As we know, Dirichlet series is regarded as a special example of the Laplace-Stieltjes transform. The Laplace-Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is an integral transform similar to the Laplace transform. For real-valued functions, it is the Laplace transform of a Stieltjes measure, however it is often defined for functions with values in a Banach space. It can be used in many fields of mathematics, such as functional analysis, and certain areas of theoretical and applied probability. In , Yu [] proved the Valiron-Knopp-Bohr formula of the associated abscissas of bounded convergence, absolute convergence, and uniform convergence of Laplace-Stieltjes.
Theorem A Suppose that Laplace-Stieltjes transforms () satisfy (), () and lim sup n→+∞ log n λ n < +∞, then where σ F u is called the abscissa of uniform convergence of F(s).
Moreover, Yu [] first introduced the maximal molecule M u (σ , G), the maximal term μ(σ , G) and the Borel line, and the order of analytic functions represented by Laplace-Stieltjes transforms convergent in the complex plane. After his works, considerable attention has been paid to the growth and value distribution of the functions represented by the Laplace-Stieltjes transform convergent in the half-plane or the whole complex plane in the field of complex analysis (see [-]).
In , Luo and Kong [] studied the following form of Laplace-Stieltjes transform: where α(x) is stated as in (), and {λ n } satisfies (),(). Set Since M(σ , F) and M u (σ , F) tend to +∞ as σ → +∞, in order to estimate the growth of F(s) more precisely, we will adapt some concepts of order, lower order, type, lower type as follows.
Definition . If Laplace-Stieltjes transform () satisfies σ F u = +∞ (the sequence {λ n } satisfies (), (), and ()) and we call F(s) of order ρ in the whole plane, where log + x = max{log x, }. If ρ ∈ (, +∞), we say that F(s) is an entire function of finite order in the whole plane. Moreover, the lower order of F(s) is defined by Remark . We say that F(s) is of the regular growth, when ρ = λ, and F(s) is of the irregular growth, when ρ = λ.
Definition . If Laplace-Stieltjes transform () satisfies σ F u = +∞ (the sequence {λ n } satisfies (), (), and ()) and is of order ρ ( < ρ < ∞), then we define the type and lower type of L-S transform F(s) as follows: Remark . The purpose of the definition of type is to compare the growth of class functions which all have the same order. For example, let f (s) = e e s , g(s) = e e s , by a simple computation, we have ρ(f ) =  = ρ(g), but T(f ) =  and T(g) = ∞. Thus, we can see that the growth of g(s) is faster than f (s) as |s| → +∞.

Results and discussion
Recently, many people studied some problems . From these references, we get the following results.
Theorem . If Laplace-Stieltjes transform () satisfies σ F u = +∞ (the sequence {λ n } satisfies (), (), and ()), and is of order ρ ( < ρ < ∞) and of type T, then Furthermore, if F(s) is of the lower order λ and the lower type τ , and λ n ∼ λ n+ and the function forms a non-decreasing function of n for n > n  , then we have From Definition ., a natural question to ask is: What happened if e σρ is replaced by e λσ in the definition of lower type when ρ = λ? We are going to consider this question.
we say that τ λ is the λ-type of F(s).
The following results are the main theorems of this paper.
Furthermore, there exists a positive integer n  such that forms a non-decreasing function of n for n > n  , then we have We denote by L β the class of all the functions F(s) of the form () which are analytic in the half-plane s < β (-∞ < β < ∞) and the sequence {λ n } satisfies () and (); and we denote by L ∞ the class of all the functions F(s) of the form () which are analytic in the half-plane s < +∞ and the sequence {λ n } satisfies (), (), and (). Thus, if -∞ < β < +∞ and F(s) ∈ L β , then F(s) ∈ L ∞ . If Laplace-Stieltjes transform () A * n =  for n ≥ k +  and A * n = , then F(s) will be called an exponential polynomial of degree k usually denoted by p k , i.e., p k (s) = λ k  exp(sy) dα(y). When we choose a suitable function α(y), the function p k (s) may be reduced to a polynomial in terms of exp(sλ i ), that is, k i= b i exp(sλ i ). For F(s) ∈ L β , -∞ < β < +∞, we denote by E n (F, β) the error in approximating the function F(s) by exponential polynomials of degree n in uniform norm as In this paper, we will further investigate the relation between E n (F, β) and the growth of an entire function defined by the L-S transform with irregular growth. It seems that this problem has never been treated before. Our main result is as follows.
Theorem . If the Laplace-Stieltjes transform F(s) ∈ L ∞ and is of lower order λ ( ≤ λ = ρ < ∞), if λ n ∼ λ n+ , then for any real number -∞ < β < +∞, we have Furthermore, there exists a positive integer n  such that forms a non-decreasing function of n for n > n  , then we have

Conclusions
From Theorems .-., we can see that the growth of Laplace-Stieltjes transforms is investigated under the assumption ρ = λ, and that some theorems about the λ-lower type τ λ , λ n , A * n , and λ are obtained. In addition, we also study the problem on the error in approximating entire functions defined by the Laplace-Stieltjes transforms. This project is a new issue of Laplace-Stieltjes transforms in the field of complex analysis. Our results are generalization and improvement of the previous conclusions given by Luo

Proofs of Theorems 2.2 and 2.3
To prove the above theorems, we require the following lemmas.

Lemma . (see [], Lemma .) If the L-S transform F(s)
where C is a constant.

.. The proof of Theorem . Since ρ > λ >  and F(s) is of the lower order λ, that is,
for any small ε( < ε < ρλ), it follows from () that there exists a constant σ  such that, for σ > σ  , and there exists a sequence {σ k } tending to +∞ such that From Lemmas . and ., we have Thus, similar to the process of (), we can easily prove Hence, this completes the proof of Theorem ..

.. The proof of Theorem
Dividing two sides of the equality in Lemma . by e ρσ and differentiating it with respect to σ , for almost all values σ > σ  , we have On the basis of the assumptions of Theorem ., taking lim sup in () when σ → +∞, from Theorem . and (), we get () easily. Similarly, dividing two sides of the equality in Lemma . by e λσ and differentiating it with respect to σ , for almost all values σ > σ  , On the basis of the assumptions of Theorem ., taking lim inf in () when σ → +∞, from Theorem . and (), we get () easily. Thus, this completes the proof of Theorem ..

The proof of Theorem 2.5
To prove this theorem, we require the following lemma.
Lemma . If the abscissa σ F u = +∞ of uniform convergence of the Laplace-Stieltjes transformation F(s) and sequence () satisfies (), (), then for any real number β, we have Proof Set For any real number β, since When n → +∞, we have b → +∞, thus we have Now, we are going to prove Theorem ..

The proof of Theorem 2.5
Let Then, for any small ε > , there exists an integer n  (ε) such that, for any n > n  (ε), Since F(s) ∈ L ∞ , thus for any constant β (-∞ < β < +∞), we have F(s) ∈ L β . For β < σ < +∞. It follows from the definitions of E n (F, β) and p n that Thus, from the definition of A * n and M u (σ , F), and by Lemma ., we have A * n ≤ M u (σ , F)e -σ λ n for any σ (β < σ < +∞). It follows from () and Lemma . that where K is a constant. Let Thus, from () and (), it follows that for n > n  (ε) By using the same argument as in Theorem ., we can easily prove that τ λ ≥ ϑ  . From the proof of Theorem ., we have that there exists a positive integer n  such that log A * n > (τ λε)e λσλ n σ for n > n  and σ > σ  . Since for any β < +∞, from the definition of E k (F, β), there exists p  ∈ n- such that And since thus for any p ∈ n- , it follows Hence from () and (), for any β < +∞ and F(s) ∈ L ∞ , we have Since e x ≥ ex for any x, so it follows Thus, for ε →  and n → +∞, from () it follows λ λn exp(-βλ), then () follows. Therefore, we complete the proof of Theorem ..