Spectral properties of orthogonal polynomials on unbounded sets
HTML articles powered by AMS MathViewer
- by T. S. Chihara PDF
- Trans. Amer. Math. Soc. 270 (1982), 623-639 Request permission
Abstract:
We consider orthogonal polynomials when the three term recurrence formula for the monic polynomials has unbounded coefficients. We obtain information relative to three questions: Under what conditions on the coefficients will the derived set of the spectrum have a finite infimum $\sigma$? If $\sigma$ is finite, when will there be at most finitely many spectral points smaller than $\sigma$; and when will the distribution function be continuous at $\sigma$?References
- W. A. Al-Salam, Characterization of certain classes of orthogonal polynomials related to elliptic functions, Ann. Mat. Pura Appl. (4) 67 (1965), 75–94. MR 178175, DOI 10.1007/BF02410805
- 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 James Wilson, A set of hypergeometric orthogonal polynomials, SIAM J. Math. Anal. 13 (1982), no. 4, 651–655. MR 661596, DOI 10.1137/0513043 O. Blumenthal, Über die entwicklung einer willkürlichen Funktion nach den Nennern des Kettenbruches für $\smallint _{ - \infty }^\infty [\phi (\xi )/(z - \xi )]\,d\xi$, Dissertation, Göttingen, 1898.
- 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, On determinate Hamburger moment problems, Pacific J. Math. 27 (1968), 475–484. MR 238038
- 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 and P. G. Nevai, Orthogonal polynomials and measures with finitely many point masses, J. Approx. Theory 35 (1982), no. 4, 370–380. MR 665989, DOI 10.1016/0021-9045(82)90025-9
- J. S. Geronimo and K. M. Case, Scattering theory and polynomials orthogonal on the real line, Trans. Amer. Math. Soc. 258 (1980), no. 2, 467–494. MR 558185, DOI 10.1090/S0002-9947-1980-0558185-4
- Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. MR 519926, DOI 10.1090/memo/0213
- Paul G. Nevai, Distribution of zeros of orthogonal polynomials, Trans. Amer. Math. Soc. 249 (1979), no. 2, 341–361. MR 525677, DOI 10.1090/S0002-9947-1979-0525677-5
- Paul G. Nevai, Orthogonal polynomials defined by a recurrence relation, Trans. Amer. Math. Soc. 250 (1979), 369–384. MR 530062, DOI 10.1090/S0002-9947-1979-0530062-6
- 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
- James A. Wilson, Some hypergeometric orthogonal polynomials, SIAM J. Math. Anal. 11 (1980), no. 4, 690–701. MR 579561, DOI 10.1137/0511064 —, Hypergeometric series recurrence relations and some new orthogonal functions, Ph.D. Thesis, Univ. of Wisconsin, Madison, 1978.
- Richard Askey and James Wilson, A set of orthogonal polynomials that generalize the Racah coefficients or $6-j$ symbols, SIAM J. Math. Anal. 10 (1979), no. 5, 1008–1016. MR 541097, DOI 10.1137/0510092
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 270 (1982), 623-639
- MSC: Primary 42C05
- DOI: https://doi.org/10.1090/S0002-9947-1982-0645334-4
- MathSciNet review: 645334