Unique solvability of an extended Stieltjes moment problem
HTML articles powered by AMS MathViewer
- by Olav Njåstad PDF
- Proc. Amer. Math. Soc. 102 (1988), 78-82 Request permission
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.References
- William B. Jones and W. J. Thron, Survey of continued fraction methods of solving moment problems and related topics, Analytic theory of continued fractions (Loen, 1981) Lecture Notes in Math., vol. 932, Springer, Berlin-New York, 1982, pp. 4–37. MR 690451
- William B. Jones, W. J. Thron, and Haakon Waadeland, A strong Stieltjes moment problem, Trans. Amer. Math. Soc. 261 (1980), no. 2, 503–528. MR 580900, DOI 10.1090/S0002-9947-1980-0580900-4
- Olav Njåstad, An extended Hamburger moment problem, Proc. Edinburgh Math. Soc. (2) 28 (1985), no. 2, 167–183. MR 806749, DOI 10.1017/S0013091500022628
- Olav Njåstad, Unique solvability of an extended Hamburger moment problem, J. Math. Anal. Appl. 124 (1987), no. 2, 502–519. MR 887006, DOI 10.1016/0022-247X(87)90011-4 —, Orthogonal rational functions with poles in a finite subset of R. II. Symposium on Orthogonal Polynomials and Their Applications, Segovia, 1986.
- Oskar Perron, Die Lehre von den Kettenbrüchen. Dritte, verbesserte und erweiterte Aufl. Bd. II. Analytisch-funktionentheoretische Kettenbrüche, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1957 (German). MR 0085349
- 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
Additional Information
- © Copyright 1988 American Mathematical Society
- 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