Nudel'man A. A.
Extension of the Stieltjes moment sequence to the left and related problems of the spectral theory of inhomogeneous string
Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 815–825
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 $d\sigma(t)$, we construct an analogous string with spectral measure $d\sigma(t)/t$.
This allows to calculate the moments of all negative orders of the measure $d\sigma(t)$.
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.
Criterion of unitary similarity of minimal passive scattering systems with a given transfer function
Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 147-156
We establish necessary and sufficient conditions under which all minimal passive scattering systems that have a given transfer operator function are unitarily equivalent. These conditions can be significantly simplified in special cases important for applications, in particular, in the case where a transfer function is rational and in a more general case where this function is pseudoextendable.
Ukr. Mat. Zh. - 1994. - 46, № 1-2. - pp. 63–75
This is a survey of М. G. Krein's papers devoted to the moment problem.