Every Stieltjes moment problem has a solution in Gel'fand-Shilov spaces

Abstract

We prove that every Stieltjes problem has a solution in Gel'fand-Shilov spaces ${\mathcal{S}}^{\beta }$ for every $\beta >1$. In other words, for an arbitrary sequence $\left\{{\mu }_n\right\}$ there exists a function $\varphi$ in the Gel'fand-Shilov space ${\mathcal{S}}^{\beta }$ with support in the positive real line whose moment ${\int }_0^{\infty }{x}^n\varphi \left(x\right)dx={\mu }_n$ for every nonnegative integer $n$.

This improves the result of A. J. Duran in 1989 very much who showed that every Stieltjes moment problem has a solution in the Schwartz space $\mathcal{S}$, since the Gel'fand-Shilov space is much a smaller subspace of the Schwartz space. Duran's result already improved the result of R. P. Boas in 1939 who showed that every Stieltjes moment problem has a solution in the class of functions of bounded variation. Our result is optimal in a sense that if $\beta \le 1$ we cannot find a solution of the Stieltjes problem for a given sequence.