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Jump distributions on $ [-1,1]$ whose orthogonal polynomials have leading coefficients with given asymptotic behavior


Author: D. S. Lubinsky
Journal: Proc. Amer. Math. Soc. 104 (1988), 516-524
MSC: Primary 42C05
DOI: https://doi.org/10.1090/S0002-9939-1988-0962822-2
MathSciNet review: 962822
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Abstract: We construct an explicit even jump distribution $ d\alpha (x)$ on $ [-1,1]$ for which $ {\gamma _n}(d\alpha )$, the leading coefficient of the $ n$th orthonormal polynomial for $ d\alpha (x)$, has any given asymptotic behavior. For example, if $ \{ {\xi _n}\} _1^\infty $ is a strictly increasing sequence with limit $ \infty $, and $ {\xi _{{3^{n + 1}}}}/{\xi _{{3^n}}} \to 1$ as $ n \to \infty $, we can ensure that $ {\lim _{n \to \infty }}{\gamma _n}(d\alpha ){2^{ - n}}/{\xi _n} = 1$. As a consequence, we have $ {\lim _{n \to \infty }}{\gamma _{n - 1}}(d\alpha )/{\gamma _n}(d\alpha ) = 1/2$; that is, $ d\alpha $ belongs to Nevai's class $ \mathcal{M}$. This positively and explicitly answers a question of Al. Magnus.


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  • [1] F. Delyon, B. Simon and B. Souillard, From power pure point to continuous spectrum in disordered systems, Ann. Inst. H. Poincaré Phys. Theor. 42 (1985), 283-309. MR 797277 (87d:35098)
  • [2] G. Freud, Orthogonal polynomials, Akademiai Kiado/Pergamon Press, Budapest, 1971.
  • [3] Ya. L. Geronimus, Orthogonal polynomials, Consultants Bureau, 1961. MR 0133643 (24:A3469)
  • [4] D. S. Lubinsky, A survey of general orthogonal polynomials for weights on finite and infinite intervals, Acta Appl. Math. 10 (1987), 237-296. MR 920673 (89h:42030)
  • [5] Al. Magnus, Toeplitz matrix techniques and convergence of complex weight Padé approximants, J. Comput. Appl. Math. 19 (1987), 23-38. MR 901209 (88i:65022)
  • [6] -, Problem section, Proceedings of the Segovia 1986 Conference on Orthogonal Polynomials.
  • [7] A. Máté and P. Nevai, Orthogonal polynomials and absolutely continuous measures, Approximation Theory IV (C. K. Chui, et al., eds.), Academic Press, New York, 1983, pp. 611-617. MR 754400 (85j:42044)
  • [8] P. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. no. 213, Amer. Math. Soc., Providence, R.I., 1979. MR 519926 (80k:42025)
  • [9] P. Nevai, Geza Freud orthogonal polynomials and Christoffel functions, a case study, J. Approx. Theory 48 (1986), 3-167. MR 862231 (88b:42032)
  • [10] G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1939, 4th ed., 1975.
  • [11] J. L. Ullman and M. F. Wyneken, Weak limits of zeros of orthogonal polynomials, Constr. Approx. 2 (1986), 339-347. MR 892160 (88e:42048)
  • [12] J. Von Neumann, Charakterieserung des Spektrums eines Integraloperators, Actualités Sci. Indust. no. 229, Hermann, Paris, 1935.
  • [13] M. F. Wyneken, Norm asymptotics of orthogonal polynomials for general measures, Constr. Approx. 4 (1988), 123-131. MR 932649 (89c:42029)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0962822-2
Keywords: Orthogonal polynomials, jump distributions, asymptotics, Nevai's class, leading coefficients, Jacobi matrices, discrete scattering
Article copyright: © Copyright 1988 American Mathematical Society

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