## A strong Stieltjes moment problem

HTML articles powered by AMS MathViewer

- by William B. Jones, W. J. Thron and Haakon Waadeland
- Trans. Amer. Math. Soc.
**261**(1980), 503-528 - DOI: https://doi.org/10.1090/S0002-9947-1980-0580900-4
- PDF | Request permission

## Abstract:

This paper is concerned with double sequences of complex numbers $C = \{ {c_n}\} _{ - \infty }^\infty$ and with formal Laurent series ${L_0}(C) = \Sigma _1^\infty - {c_{ - m}}{z^m}$ and ${L_\infty }(C) = \Sigma _0^\infty {c_m}{z^{ - m}}$ generated by them. We investigate the following related problems: (1) Does there exist a holomorphic function having ${L_0}(C)$ and ${L_\infty }(C)$ as asymptotic expansions at $z = 0$ and $z = \infty$, respectively? (2) Does there exist a real-valued bounded, monotonically increasing function $\psi (t)$ with infinitely many points of increase on $[0, \infty )$ such that, for every integer*n*, ${c_n} = \int _0^\infty {{{( - t)}^n} d\psi (t)}$? The latter problem is called the strong Stieltjes moment problem. We also consider a modified moment problem in which the function $\psi (t)$ has at most a finite number of points of increase. Our approach is made through the study of a special class of continued fractions (called positive

*T*-fractions) which correspond to ${L_0}(C)$ at $z = 0$ and ${L_\infty }(C)$ at $z = \infty$. Necessary and sufficient conditions are given for the existence of these corresponding continued fractions. It is further shown that the even and odd parts of these continued fractions always converge to holomorphic functions which have ${L_0}(C)$ and ${L_\infty }(C)$ as asymptotic expansions. Moreover, these holomorphic functions are shown to be represented by Stieltjes integral transforms whose distributions ${\psi ^{(0)}}(t)$ and ${\psi ^{(1)}}(t)$ solve the strong Stieltjes moment problem. Necessary and sufficient conditions are given for the existence of a solution to the strong Stieltjes moment problem. This moment problem is shown to have a unique solution if and only if the related continued fraction is convergent. Finally it is shown that the modified moment problem has a unique solution if and only if there exists a terminating positive

*T*-fraction that corresponds to both ${L_0}(C)$ and ${L_\infty }(C)$. References are given to other moment problems and to investigations in which negative, as well as positive, moments have been used.

## References

- Zakkula Govindarajulu,
*Recurrence relations for the inverse moments of the positive binomial variable*, J. Amer. Statist. Assoc.**58**(1963), 468–473. MR**149583** - Peter Henrici,
*Applied and computational complex analysis*, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Volume 1: Power series—integration—conformal mapping—location of zeros. MR**0372162**
T. Jefferson, - William B. Jones,
*Multiple-point Padé tables*, Padé and rational approximation (Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976) Academic Press, New York, 1977, pp. 163–171. MR**0613840** - William B. Jones and W. J. Thorn,
*Further properties of $T$-fractions*, Math. Ann.**166**(1966), 106–118. MR**200425**, DOI 10.1007/BF01361442 - William B. Jones and W. J. Thron,
*Two-point Padé tables and $T$-fractions*, Bull. Amer. Math. Soc.**83**(1977), no. 3, 388–390. MR**447543**, DOI 10.1090/S0002-9904-1977-14284-5 - D. G. Kabe,
*Inverse moments of discrete distributions*, Canad. J. Statist.**4**(1976), no. 1, 133–141 (English, with French summary). MR**448656**, DOI 10.2307/3315269 - J. H. McCabe,
*A formal extension of the Padé table to include two point Padé quotionts*, J. Inst. Math. Appl.**15**(1975), 363–372. MR**381246** - J. H. McCabe and J. A. Murphy,
*Continued fractions which correspond to power series expansions at two points*, J. Inst. Math. Appl.**17**(1976), no. 2, 233–247. MR**422628** - W. Mendenhall and E. H. Lehman Jr.,
*An approximation to the negative moments of the positive binomial useful in life testing*, Technometrics**2**(1960), 227–242. MR**114279**, DOI 10.2307/1266547 - I. P. Natanson,
*Theory of functions of a real variable*, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron with the collaboration of Edwin Hewitt. MR**0067952** - Oskar Perron,
*Die Lehre von den Kettenbrüchen*, Chelsea Publishing Co., New York, N. Y., 1950 (German). 2d ed. MR**0037384** - T.-J. Stieltjes,
*Recherches sur les fractions continues*, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys.**8**(1894), no. 4, J1–J122 (French). MR**1508159** - D. L. Thomas,
*Reciprocal moments of linear combinations of exponential variates*, J. Amer. Statist. Assoc.**71**(1976), no. 354, 506–512. MR**403015**
W. J. Thron, - W. J. Thron,
*Two-point Padé tables, $T$-fractions and sequences of Schur*, Padé and rational approximation (Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976) Academic Press, New York, 1977, pp. 215–226. MR**0507914** - Haakon Waadeland,
*On $T$-fractions of certain functions with a first order pole at the point of infinity*, Norske Vid. Selsk. Forh. (Trondheim)**40**(1967), 1–6. MR**233968** - H. S. Wall,
*Analytic Theory of Continued Fractions*, D. Van Nostrand Co., Inc., New York, N. Y., 1948. MR**0025596**
D. V. Widder,

*Some additional properties of T-fractions*, Ph. D. Thesis, University of Colorado, Boulder, Colorado, 1969.

*Some properties of continued fractions*$1\, + \,{d_0}z\, + \,K(z/1\, + \,{d_n}z)$, Bull. Amer. Math. Soc.

**54**(1948), 112-120.

*The Laplace transform*, Princeton Univ. Press, Princeton, N. J., 1948.

## Bibliographic Information

- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**261**(1980), 503-528 - MSC: Primary 30E05; Secondary 30B70
- DOI: https://doi.org/10.1090/S0002-9947-1980-0580900-4
- MathSciNet review: 580900