A Strong Stieltjes Moment Problem

Abstract

This paper is concerned with double sequences of complex numbers $C = \{ {c_n}\} _{ - \infty }^\infty$ and with formal Laurent series ${L_0}(C) = \Sigma _1^\infty - {c_{ - m}}{z^m}$ and ${L_\infty }(C) = \Sigma _0^\infty {c_m}{z^{ - m}}$ generated by them. We investigate the following related problems: (1) Does there exist a holomorphic function having ${L_0}(C)$ and ${L_\infty }(C)$ as asymptotic expansions at $z = 0$ and $z = \infty$, respectively? (2) Does there exist a real-valued bounded, monotonically increasing function $\psi (t)$ with infinitely many points of increase on $[0, \infty )$ such that, for every integer n, ${c_n} = \int _0^\infty {{{( - t)}^n} d\psi (t)}$? The latter problem is called the strong Stieltjes moment problem. We also consider a modified moment problem in which the function $\psi (t)$ has at most a finite number of points of increase. Our approach is made through the study of a special class of continued fractions (called positive T-fractions) which correspond to ${L_0}(C)$ at $z = 0$ and ${L_\infty }(C)$ at $z = \infty$. Necessary and sufficient conditions are given for the existence of these corresponding continued fractions. It is further shown that the even and odd parts of these continued fractions always converge to holomorphic functions which have ${L_0}(C)$ and ${L_\infty }(C)$ as asymptotic expansions. Moreover, these holomorphic functions are shown to be represented by Stieltjes integral transforms whose distributions ${\psi ^{(0)}}(t)$ and ${\psi ^{(1)}}(t)$ solve the strong Stieltjes moment problem. Necessary and sufficient conditions are given for the existence of a solution to the strong Stieltjes moment problem. This moment problem is shown to have a unique solution if and only if the related continued fraction is convergent. Finally it is shown that the modified moment problem has a unique solution if and only if there exists a terminating positive T-fraction that corresponds to both ${L_0}(C)$ and ${L_\infty }(C)$. References are given to other moment problems and to investigations in which negative, as well as positive, moments have been used.