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

概要

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We prove that every Stieltjes problem has a solution in Gelfand-Shilov spaces \mathscr{S}β for every β>1. In other words, for an arbitrary sequence {μn} there exists a function \varphi in the Gelfand-Shilov space \mathscr{S}β with support in the positive real line whose moment \displaystyle ∈t0∞xn\varphi(x)dx=μn 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 \mathscr{S}, since the Gelfand-Shilov space is much a smaller subspace of the Schwartz space. Durans 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 β≤ 1 we cannot find a solution of the Stieltjes problem for a given sequence.
社団法人日本数学会の論文
2003-10-01