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Rational Approximation and Interpolation pp 524–528Cite as

Orthogonal polynomials for general measures-I

  • Orthogonal Polynomials
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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1105))

Abstract

Associated with a unit Borel measure in the complex plane, α, whose support K(α) is compact and contains infinitely many points is a family of orthonormal polynomials {φn(z)}, n=0,1,… . The family of potentials ωn(z)=1 / n log|φn(z)| will be studied. Conditions have previously been found which insure that ωn(z) behaves like the Green's function of O(K(α)), the unbounded component of the complement of K(α). We study the behavior of ωn(z) when these conditions are not satisfied.

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References

  1. Landkof, N. S., Foundations of Modern Potential Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1972.

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  2. Nguyen, Thanh Van, Familles de polynômes ponctuellement bonnees, Annals Polonici Mathematici, XXXI (1975) 83–90.

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  3. Ullman, J. L., On the regular behavior of orthogonal polynomials, Proc. London Math. Soc. (3) 24 (1972) 119–148.

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  4. Ullman, J. L., A survey of exterior asymptotics for orthogonal polynomials, Proceedings of the N.A.T.O. Advanced Study Institute, St. Johns, Nfld, 1983.

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  5. Ullman, J. L., Orthogonal polynomials for general measures, II. In preparation.

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  6. Widom, Harold, Polynomials associated with measures in the complex plane, Journal of Math. and Mech., Vol. 16, No. 9 (1967) 997–1013.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 48109-1003, Ann Arbor, Michigan

    Joseph L. Ullman

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  1. Joseph L. Ullman
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Peter Russell Graves-Morris Edward B. Saff Richard S. Varga

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© 1984 Springer-Verlag

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Ullman, J.L. (1984). Orthogonal polynomials for general measures-I. In: Graves-Morris, P.R., Saff, E.B., Varga, R.S. (eds) Rational Approximation and Interpolation. Lecture Notes in Mathematics, vol 1105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072438

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  • DOI: https://doi.org/10.1007/BFb0072438

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13899-0

  • Online ISBN: 978-3-540-39113-5

  • eBook Packages: Springer Book Archive

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