Unique solvability of an extended Stieltjes moment problem

Abstract:

Let ${a_1},{a_2}, \ldots ,{a_p}$ be given real numbers ordered by size, and let $[\alpha ,\beta ]$ be a real interval disjoint from the set $\{ {a_1},{a_2}, \ldots ,{a_p}\}$. Let $\{ c_j^{(i)}:j = 1,2, \ldots \}$, be sequences of real numbers and ${c_0}$ be a real number. The extended Stieltjes moment problem is to find a distribution function $\psi$ with all its points of increase in $[\alpha ,\beta ]$ such that \[ \int _\alpha ^\beta {d\psi (t) = {c_0},\quad \int _\alpha ^\beta {\frac {{d\psi (t)}}{{{{(t - {a_i})}^j}}} = c_j^{(i)},\quad i = 1, \ldots ,p,\;j = 1,2, \ldots .} } \] Necessary and sufficient conditions for the existence of a unique solution of the problem are given. Orthogonal $R$-functions and Gaussian quadrature formulas play important roles in the proof.