On Banach Spaces and Fréchet Spaces of Laplace–Stieltjes Integrals

We investigate the spaces of Laplace–Stieltjes integrals $I\left(\sigma \right)={\int}_0^{\infty }f(x){e}^{x\sigma} dF(x),$ σ ∈ ℝ, F is a nonnegative nondecreasing unbounded function right continuous on [0, +∞), and f is a real-valued function on [0, +∞). This integral is a generalization of the Dirichlet series $D\left(\sigma \right)={\sum}_{n=1}^{\infty }{d}_n{e}^{\uplambda_n\sigma }$ with nonnegative exponents λ_n increasing to +∞ if $F(x)=n(x)={\sum}_{\lambda_n\le x}1,$ and f (x) = d_n for x = λ_n and f (x) = 0 for x ≠ λ_n. For a positive continuous function h on [0, +∞) that increases to +∞, by LS_h we denote a class of integrals I such that |f(x)| exp {xh(x)} → 0 as x →  + ∞ and define ‖I‖_h = sup {|f(x)| exp {xh(x)} : x ≥ 0}. We prove that if F ∈ V and ln F(x) = o(x) as x → +∞, then (LS_h, ‖⋅‖_h) is a nonuniformly convex Banach space. Some other properties of the space LS_h and its dual space are also studied. As a consequence, we obtain results for the Banach spaces of Laplace–Stieltjes integrals of finite generalized order. Some results are refined in the case where I (σ) = D(σ). In addition, for fixed ϱ < +∞, we assume that ${\overline{S}}_{\upvarrho}$ is a class of entire Dirichlet series D (σ) such that their generalized order ${\upvarrho}_{\upalpha, \beta}\left[D\right]:= \underset{\upsigma \to +\infty }{\lim \kern0.5em \sup}\frac{\alpha \left(\ln M\left(\sigma, D\right)\right)}{\beta \left(\sigma \right)}\le \upvarrho,$ where $M\left(\upsigma, D\right)={\sum}_{n=1}^{\infty}\left|{d}_n\right|{e}^{\upsigma {\lambda}_n}$ and the functions α and β are positive, continuous on [x₀, +∞), and increasing to +∞. Further, for q ∈ ℕ, let

$$ {\left\Vert D\right\Vert}_{\upvarrho; q}=\sum \limits_{n=1}^{\infty}\left|{d}_n\right|\exp \left\{{\lambda}_n{\beta}^{-1}\left(\frac{\alpha \left({\lambda}_n\right)}{\upvarrho +1/q}\right)\right\},\kern2em d\left({D}_1,{D}_2\right)=\sum \limits_{q=1}^{\infty}\frac{1}{2^q}\frac{{\left\Vert {D}_1-{D}_2\right\Vert}_{\upvarrho; q}}{1+{\left\Vert {D}_1-{D}_2\right\Vert}_{\upvarrho; q}}. $$

The space with the metric d is denoted by ${\overline{S}}_{\upvarrho, d}$ is a Fréchet space under certain conditions imposed on the functions α and β and the sequence (λ_n).