On a Banach space of Laplace-Stieltjes integrals

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On a Banach space of Laplace-Stieltjes integrals

Journal Title: Математичні Студії - Year 2017, Vol 48, Issue 2

Abstract

Let Ω be class of positive unbounded functions Φ on (−∞,+∞) such that the derivative Φ′ is positive, continuously differentiable and increasing to +∞ on (−∞,+∞), φ be the inverse function to Φ′, and Ψ(x)=x−Φ(x)Φ′(x) be the function associated with Φ in the sense of Newton. Let F be nonnegative nondecreasing unbounded continuous on the right function on [0,+∞) and f be a real-value function on [0,+∞). By LSΦ(F) we denote the class of integrals I(σ)=∫∞0f(x)exσdF(x), convergent for all σ∈R such that |f(x)|exp{xΨ(φ(x))}→0 as x→+∞. Put ∥I∥Φ:=sup{|f(x)|exp{xΨ(φ(x))}:x≥0}. It is proved that if lnF(x)=o(x) as x→+∞ then (LSΦ(F),∥⋅∥Φ) is a Banach space and it is studied its properties.

Authors and Affiliations

Myroslav Sheremeta, Marian Dobushovskyy, Andriy Kuryliak

Keywords

Laplace-Stieltjes integral; Dirichlet series; Banach space

EP ID EP302521
DOI 10.15330/ms.48.2.143-149
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