Abstract
This paper is concerned with indeterminate classical Stieltjes moment problems (CSMPs) and indeterminate strong Stieltjes moment problems (SSMPs). We are particularly interested in what we call the natural solutions to each type of moment problem. Known results are summarized and we compare Stieltjes’ results on indeterminate CSMPs to analogous results for a special class of indeterminate SSMPs. For the indeterminate CSMP each natural solution is a step function whose points of increase have a limit point at ∞. Each natural solution to an indeterminate SSMP is a step function whose points of increase have limit points only at 0 and/or ∞. Every convex linear combination of the natural solutions to an indeterminate Stieltjes moment problem (classical and strong) is also a solution to that problem. It is shown that there exist indeterminate Stieltjes moment problems (both classical and strong) which have step function solutions which are not convex linear combinations of the natural solutions. Finally, it is shown how to construct new moment problems and their solutions from known moment problems and their solutions so that the moments of both problems are related by “shifts” and “flips.”
Research supported in part by the National science Foundation under Grant No. DMS-9302584
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Catherine Bonan-Hamada, William B. Jones and W. J. Thron, A class of indeterminate strong Stieltjes moment problems with discrete distributions, submitted.
John B. Conway, Functions of One Complex Variable, Springer-Verlag, New York, 1973.
Einar Hille, Analytic Function Theory, Vol. I, Blaisdell, New York, 1959.
William B. Jones and W. J. Thron, Continued Fractions: Analytic Theory and Applications, in Encyclopedia of Mathematics and Its Applications, Vol. 11, Addison-Wesley, Reading, Mass., 1980. Distributed now by the Cambridge University Press.
William B. Jones, W. J. Thron and Haakon Waadeland, A strong Stieltjes moment problem, Trans. Amer. Math Soc. 261 (1980), 503–528.
Olav Njåstad, personal communication, 1993.
T. J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse (1894), J. 1–122; 9 (1894), A, 1–47; Oeuvres, Vol. 2, pp. 402–566. Also published in Mémoires présentés par divers savants a l’Académie de sciences de l’Institut National de France, Vol. 33, pp. 1–196. This paper and an English translation are now also available in Oeuvres Complétes - Collected Papers (G. van Dijk, ed.), Springer-Verlag, 1993, vol.2, pp. 406–570; pp. 609–745 (English).
W. J. Thron, Introduction to the Theory of a Complex Variable, Wiley, New York, 1953.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Bonan-Hamada, C.M., Jones, W.B., Thron, W.J. (1994). Observations on Indeterminate Stieltjes Moment Problems. In: Cuyt, A. (eds) Nonlinear Numerical Methods and Rational Approximation II. Mathematics and Its Applications, vol 296. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0970-3_30
Download citation
DOI: https://doi.org/10.1007/978-94-011-0970-3_30
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-4420-2
Online ISBN: 978-94-011-0970-3
eBook Packages: Springer Book Archive