We prove that every Stieltjes problem has a solution in Gel'fand-Shilov spaces for every . In other words, for an arbitrary sequence there exists a function in the Gel'fand-Shilov space with support in the positive real line whose moment for every nonnegative integer .
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 , 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 we cannot find a solution of the Stieltjes problem for a given sequence.
"Every Stieltjes moment problem has a solution in Gel'fand-Shilov spaces." J. Math. Soc. Japan 55 (4) 909 - 913, October, 2003. https://doi.org/10.2969/jmsj/1191418755