The Stieltjes moments problem for rapidly decreasing functions

Abstract:

We prove the following result: If ${\left ( {{a_n}} \right )_n}$ is a sequence of complex numbers, then there exists a ${\mathcal {C}^\infty }$-function $f$ such that $f$ and all its derivatives are rapidly decreasing functions, $f\left ( t \right ) = 0$ for $t < 0$ and $\int _0^{ + \infty } {{t^n}f\left ( t \right )dt = {a_n}}$. We extend this result for a generalized Stieltjes moments problem. Also, we characterize the ${C^\infty }$-functions $f$ in $\left ( {0, + \infty } \right )$ such that $f$ and all its derivatives are rapidly decreasing functions in $\left ( {0, + \infty } \right )$ and with null moments.