Extension of the Stieltjes moment sequence to the left and related problems of the spectral theory of inhomogeneous string

Abstract

For an inhomogeneous string with known mass distribution (the total mass is assumed to be infinite), known finite length, and unknown spectral measure dσ(t), we construct an analogous string with spectral measure dσ(t)/t. This enables one to determine the moments of all non-negative orders for the measure dσ(t). The mechanical interpretation of Stieltjes’ investigation of the problem of moments proposed by Krein enables one to solve the problem of finding the moments of negative orders for the Stieltjes moment sequence that has a unique solution. This problem is equivalent to the problem of determining the asymptotic behavior of the associated Stieltjes function near zero on the basis of its known asymptotic behavior at infinity.