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.