We give a sufficient condition for a non-negative random variable $X$ to be of Pareto type by investigating the Laplace-Stieltjes transform of the cumulative distribution function. We focus on the relation between the singularity at the real point of the axis of convergence and the asymptotic decay of the tail probability. For the proof of our theorems, we apply Graham-Vaaler’s complex Tauberian theorem. As an application of our theorems, we consider the asymptotic decay of the stationary distribution...

The aim of this paper is to derive by elementary means a theorem on the representation of certain distributions in the form of a Fourier integral. The approach chosen was found suitable especially for students of post-graduate courses at technical universities, where it is in some situations necessary to restrict a little the extent of the mathematical theory when concentrating on a technical problem.

The harmonic Cesàro operator is defined for a function f in $L^p\left(ℝ\right)$ for some 1 ≤ p < ∞ by setting $\left(f\right)\left(x\right):={\int }_x^{\infty }(f\left(u\right)/u)du$ for x > 0 and $\left(f\right)\left(x\right):=-{\int }_{-\infty }^x(f\left(u\right)/u)du$ for x < 0; the harmonic Copson operator ℂ* is defined for a function f in $L{¹}_{loc}\left(ℝ\right)$ by setting $*\left(f\right)\left(x\right):=(1/x){\int }^{x₀}f\left(u\right)du$ for x ≠ 0. The notation indicates that ℂ and ℂ* are adjoint operators in a certain sense. We present rigorous proofs of the following two commuting relations: (i) If $f\in L^p\left(ℝ\right)$ for some 1 ≤ p ≤ 2, then ${\left(\left(f\right)\right)}^{\wedge }\left(t\right)=*\left(f̂\right)\left(t\right)$ a.e., where f̂ denotes the Fourier transform of f. (ii) If $f\in L^p\left(ℝ\right)$ for some 1 < p ≤ 2, then ${(*\left(f\right))}^{\wedge }\left(t\right)=\left(f̂\right)\left(t\right)$ a.e. As...

This work studies conditions that insure the existence of weak boundary values for solutions of a complex, planar, smooth vector field $L$. Applications to the F. and M. Riesz property for vector fields are discussed.

Nous étudions d’abord la transformation de Fourier sur les espaces ${\ell }^p(L^{p^{\text{'}}})$ qui sont formés de fonctions appartenant localement à $L^{p^{\text{'}}}$ et se comportant à l’infini comme des éléments de ${\ell }^p$. Si $1\le p,p^{\text{'}}\le 2$, les transformées de Fourier des éléments de ${\ell }^p(L^{p^{\text{'}}})$ appartiennent à ${\ell }^{q^{\text{'}}}(L^q)$. Dans les autres cas, nous donnons quelques résultats partiels.Nous montrons ensuite que ${\ell }^2(L^1)$ est le plus grand espace vectoriel solide de fonctions mesurables sur lequel la transformation de Fourier puisse se définir par prolongement par continuité.