Hamburger moment problems and orthogonal polynomials
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- by T. S. Chihara PDF
- Trans. Amer. Math. Soc. 315 (1989), 189-203 Request permission
Abstract:
We consider a sequence of orthogonal polynomials given by the classical three term recurrence relation. We address the problem of deciding the determinacy or indeterminacy of the associated Hamburger moment problem on the basis of the behavior of the coefficients in the three term recurrence relation. Comparisons are made with other criteria in the literature. The efficacy of the criteria obtained is illustrated by application to many specific examples of orthogonal polynomials.References
-
N. I. Ahiezer, The classical moment problem, Hafner, New York, 1965.
- W. A. Al-Salam and L. Carlitz, Some orthogonal $q$-polynomials, Math. Nachr. 30 (1965), 47–61. MR 197804, DOI 10.1002/mana.19650300105
- Richard Askey and Mourad Ismail, Recurrence relations, continued fractions, and orthogonal polynomials, Mem. Amer. Math. Soc. 49 (1984), no. 300, iv+108. MR 743545, DOI 10.1090/memo/0300 T. Carleman, Sur les équations intégrales singulières à noyau réel et symmétrique, Uppsala Universitets Arsskrift, 1923, 228 pp.
- T. S. Chihara, Chain sequences and orthogonal polynomials, Trans. Amer. Math. Soc. 104 (1962), 1–16. MR 138933, DOI 10.1090/S0002-9947-1962-0138933-7
- T. S. Chihara, On recursively defined orthogonal polynomials, Proc. Amer. Math. Soc. 16 (1965), 702–710. MR 179398, DOI 10.1090/S0002-9939-1965-0179398-7
- T. S. Chihara, Convergent sequences of orthogonal polynomials, J. Math. Anal. Appl. 38 (1972), 335–347. MR 308673, DOI 10.1016/0022-247X(72)90092-3
- T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
- T. S. Chihara, Orthogonal polynomials whose distribution functions have finite point spectra, SIAM J. Math. Anal. 11 (1980), no. 2, 358–364. MR 559875, DOI 10.1137/0511033
- T. S. Chihara, Indeterminate symmetric moment problems, J. Math. Anal. Appl. 85 (1982), no. 2, 331–346. MR 649179, DOI 10.1016/0022-247X(82)90005-1
- T. S. Chihara, Spectral properties of orthogonal polynomials on unbounded sets, Trans. Amer. Math. Soc. 270 (1982), no. 2, 623–639. MR 645334, DOI 10.1090/S0002-9947-1982-0645334-4
- Joseph J. Dennis and H. S. Wall, The limit-circle case for a positive definite $J$-fraction, Duke Math. J. 12 (1945), 255–273. MR 13436
- Jacob Sherman, On the numerators of the convergents of the Stieltjes continued fractions, Trans. Amer. Math. Soc. 35 (1933), no. 1, 64–87. MR 1501672, DOI 10.1090/S0002-9947-1933-1501672-3
- J. A. Shohat and J. D. Tamarkin, The Problem of Moments, American Mathematical Society Mathematical Surveys, Vol. I, American Mathematical Society, New York, 1943. MR 0008438
- H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Co., Inc., New York, N. Y., 1948. MR 0025596
- H. S. Wall and Marion Wetzel, Contributions to the analytic theory of $J$-fractions, Tran. Amer. Math. Soc. 55 (1944), 373–392. MR 0011339, DOI 10.1090/S0002-9947-1944-0011339-6
- James A. Wilson, Some hypergeometric orthogonal polynomials, SIAM J. Math. Anal. 11 (1980), no. 4, 690–701. MR 579561, DOI 10.1137/0511064
- Arthur Wouk, Difference equations and $J$-matrices, Duke Math. J. 20 (1953), 141–159. MR 58111
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 315 (1989), 189-203
- MSC: Primary 42C05; Secondary 30E05, 33A65, 44A60
- DOI: https://doi.org/10.1090/S0002-9947-1989-0986686-1
- MathSciNet review: 986686