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Orthogonal polynomials and jump modifications

  • M. A. Cachafeiro
  • F. Marcellán
Contributors
Part of the Lecture Notes in Mathematics book series (LNM, volume 1329)

Abstract

Let \(\left\{ {\hat P_n \left( {z,d\alpha } \right)} \right\}\) be a system of orthonormal polynomials on the unit circle U with respect to a measure α and let \(\left\{ {\hat P_n \left( {z,d\beta } \right)} \right\}\) be a system of orthonormal polynomials on U with respect to β (β=α+εδ(t), where t≠U, ε>0, and δ(t) the unit measure supported at z=t). If (en (α))−1/2 and (en (β))−1/2 are their leading coefficients, in this paper we present some properties of the difference en (β)-en(α) and we show that both se quences en (α) and en (β) have the same limit by using the recurrence re lation.

Keywords

Unit Circle Orthogonal Polynomial Approximation Theory Unit Measure Consultant Bureau 
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References

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    CACHAFEIRO, M.A.; MARCELLAN, F. "Polinomios ortogonales y medidas singulares sobre curvas". Comunicación a las XI Jornadas Hispano-Lusas de Matemáticas. Badajoz, 1986.Google Scholar
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    FREUD, G. "Orthogonal Polynomials". Pergamon Press. New York, 1971.zbMATHGoogle Scholar
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    GERONIMUS, L. "Orthogonal Polynomials". Consultants Bureau. New York, 1961.zbMATHGoogle Scholar
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    MATE, A.; NEVAI, P. "Remarks on E. A. Rahmanov's Paper: "On the asymptotics of the ratio of Orthogonal Polynomials"". Journal of Approximation Theory. 36 (1982) pp. 64–72.MathSciNetCrossRefzbMATHGoogle Scholar
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    RAHMANOV, E.A. "On the asymptotics of the ratio of Orthogonal Polynomials II". Math. U.S.S.R. Sb. 46 (1983) pp. 105–117.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • M. A. Cachafeiro
    • 1
  • F. Marcellán
    • 2
  1. 1.Departamento de Matemática AplicadaE.T.S. Ingenieros IndustrialesVigoSpain
  2. 2.Departamento de Matemática AplicadaE.T.S. Ingenieros IndustrialesMadridSpain

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