Extension of the Stieltjes moment sequence to the left and related problems of the spectral theory of inhomogeneous string - Nudel'man, A. A. (Ukrains’kyi Matematychnyi Zhurnal, 2007)
For a nonhomogeneous string with the known mass distribution (the full mass is assumed to be infinite), the known finite length, and the unknown spectral measure $dσ(t)$, we construct an analogous string with spectral measure $dσ(t)/t$. This allows to calculate the moments of all negative orders of the measure $dσ(t)$. The mechanical interpretation of the Stieltjes investigations on the moment problem proposed by M. G. Krein enables one to solve the following problem: for given Stieltjes moment sequence with unique solution, calculate the moments of negative orders. This problem is equivalent to the following one: establish the asymptotic behavior of the associate Stieltjes function near zero if its asymptotic behavior near infinity is given.