Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1988-0915720-4
MathSciNet review: 915720
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] W. B. Jones and W. J. Thron, Survey of continued fraction methods of solving moment problems and related topics, Analytic Theory of Continued Fractions (Proceedings Loen, 1981), Lecture Notes in Math., vol. 932, Springer, Berlin, 1982, pp. 4-37. MR 690451 (84k:44015)
  • [2] W. B. Jones, W. J. Thron, and H. Waadeland, A strong Stieltjes moment problem, Trans. Amer. Math. Soc. 261 (1980), 503-528. MR 580900 (81j:30055)
  • [3] O. Njåstad, An extended Hamburger moment problem, Proc. Edinburgh Math. Soc. 28 (1985), 167-183. MR 806749 (86k:44011)
  • [4] -, Unique solvability of an extended Hamburger moment problem, J. Math. Anal. Appl. 124 (1987), 502-519. MR 887006 (88g:44006)
  • [5] -, Orthogonal rational functions with poles in a finite subset of R. II. Symposium on Orthogonal Polynomials and Their Applications, Segovia, 1986.
  • [6] O. Perron, Die Lehre von den Kettenbrüchen, 3. Auflage, Band 2, Teubner, Stuttgart, 1957. MR 0085349 (19:25c)
  • [7] T. J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Math. 8 (1894), 1-122. MR 1508159

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30E05,, 42C05

Retrieve articles in all journals with MSC: 30E05,, 42C05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0915720-4
Keywords: Moment problems, orthogonal $ R$-functions
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society