Questions tagged [stieltjes-integral]

What is the procedure to find the weight function $p(t)$ of a Stieltjes integral $F(x)=\int_0^\infty p(t)/(1+xt)dt$ such that $F(x)=\sum_{n=0}^\infty a_n (-x)^n$ and $a_n=\int_0^\infty t^n p(t)dt$? For example, I know that $F(x)=log(1+x)/x$ is a…

A function $f(x)$ is completely monotonic if $(-1)^nf^{(n)}(x) \geq 0$ for all $n \geq 0$ and all $x > 0$. These functions are characterized by Bernstein's theorem and are well studied in Widder's book on Laplace transforms where $g(t) =…

In Riemann-Stieltjes integral, $ \int_{0}^{\infty} f(x) w(x)dx$ why the weight function should be nontrivial when the interval is $[0,\infty)$?