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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Unique solvability of an extended Stieltjes moment problem

Author: Olav Njåstad
Journal: Proc. Amer. Math. Soc. 102 (1988), 78-82
MSC: Primary 30E05,; Secondary 42C05
MathSciNet review: 915720
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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

$\displaystyle \int_\alpha ^\beta {d\psi (t) = {c_0},\quad \int_\alpha ^\beta {\... ...{{{(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.

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Keywords: Moment problems, orthogonal $ R$-functions
Article copyright: © Copyright 1988 American Mathematical Society