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Zeros of Sobolev Orthogonal Polynomials via Muckenhoupt Inequality with Three Measures

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Abstract

A good strategy in order to obtain the asymptotic behavior of Sobolev orthogonal polynomials is to prove that the multiplication operator is bounded in the appropriate Sobolev space, which implies the boundedness of their zeros. In this paper we obtain a very simple characterization of the boundedness of the multiplication operator, by proving a generalization of the Muckenhoupt inequality with two measures to three. These results are obtained for a large class of measures which includes the most usual examples, for instance, every Jacobi weight (and even every generalized Jacobi weight) with any finite amount of Dirac deltas.

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References

  1. Alvarez, V., Pestana, D., Rodríguez, J.M., Romera, E.: Weighted Sobolev spaces on curves. J. Approx. Theory 119, 41–85 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cachafeiro, A., Marcellán, F.: The characterization of the quasi-typical extension of an inner product. J. Approx. Theory 62, 235–242 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Castro, M., Durán, A.J.: Boundedness properties for Sobolev inner products. J. Approx. Theory 122, 97–111 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Colorado, E., Pestana, D., Rodríguez, J.M., Romera, E.: Muckenhoupt inequality with three measures and Sobolev orthogonal polynomials. J. Math. Anal. Appl. 407, 369–386 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fefferman, C.L.: The uncertainty principle. Bull. Am. Math. Soc. 9(2), 129–206 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  6. García-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies, vol. 116. North-Holland, Amsterdam (1985)

    Book  MATH  Google Scholar 

  7. Gautschi, W.: On the computation of special Sobolev-type orthogonal polynomials. Ann. Numer. Math. 4, 329–342 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Gautschi, W., Kuijlaars, A.B.J.: Zeros and critical points of Sobolev orthogonal polynomials. J. Approx. Theory 91, 117–137 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Heisenberg, W.: On the perceptual content of quantum theoretical kinematics and mechanics. Z. Phys. 43, 172–198 (1927). English translation by John A. Wheeler and Wojciech Zurek, eds. Quantum Theory and Measurement, Princeton University Press, Princeton, 1983, pp. 62–84

    Article  MATH  Google Scholar 

  10. Kufner, A., Opic, B.: How to define reasonably weighted Sobolev spaces. Comment. Math. Univ. Carol. 25(3), 537–554 (1984)

    MathSciNet  MATH  Google Scholar 

  11. López Lagomasino, G., Pérez Izquierdo, I., Pijeira, H.: Asymptotic of extremal polynomials in the complex plane. J. Approx. Theory 137, 226–237 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. López Lagomasino, G., Pijeira, H.: Zero location and n-th root asymptotics of Sobolev orthogonal polynomials. J. Approx. Theory 99, 30–43 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Maz’ja, V.G.: Sobolev Spaces. Springer, New York (1985)

    Book  Google Scholar 

  14. Muckenhoupt, B.: Hardy’s inequality with weights. Stud. Math. 44, 31–38 (1972)

    MathSciNet  MATH  Google Scholar 

  15. Portilla, A., Quintana, Y., Rodríguez, J.M., Tourís, E.: Weierstrass’ theorem with weights. J. Approx. Theory 127, 83–107 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Portilla, A., Quintana, Y., Rodríguez, J.M., Tourís, E.: Zero location and asymptotic behavior for extremal polynomials with non-diagonal Sobolev norms. J. Approx. Theory 162, 2225–2242 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Portilla, A., Quintana, Y., Rodríguez, J.M., Tourís, E.: Concerning asymptotic behavior for extremal polynomials associated to non-diagonal Sobolev norms. J. Funct. Spaces Appl. 2013, 628031 (2013), 11 pages

    Article  MATH  Google Scholar 

  18. Portilla, A., Rodríguez, J.M., Tourís, E.: The multiplication operator, zero location and asymptotic for non-diagonal Sobolev norms. Acta Appl. Math. 111, 205–218 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Recchioni, M.C.: Quadratically convergent method for simultaneously approaching the roots of polynomial solutions of a class of differential equations: application to orthogonal polynomials. In memory of W. Gross. Numer. Algorithms 28, 285–308 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rodríguez, J.M., Alvarez, V., Romera, E., Pestana, D.: Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials I. Acta Appl. Math. 80, 273–308 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rodríguez, J.M., Alvarez, V., Romera, E., Pestana, D.: Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials II. Approx. Theory Appl. 18(2), 1–32 (2002)

    MathSciNet  MATH  Google Scholar 

  22. Rodríguez, J.M., Alvarez, V., Romera, E., Pestana, D.: Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials: a survey. Electron. Trans. Numer. Anal. 24, 88–93 (2006)

    MathSciNet  MATH  Google Scholar 

  23. Rodríguez, J.M.: The multiplication operator in Sobolev spaces with respect to measures. J. Approx. Theory 109, 157–197 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rodríguez, J.M.: A simple characterization of weighted Sobolev spaces with bounded multiplication operator. J. Approx. Theory 153, 53–72 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rodríguez, J.M., Sigarreta, J.M.: Sobolev spaces with respect to measures in curves and zeros of Sobolev orthogonal polynomials. Acta Appl. Math. 104, 325–353 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Grundlehren der mathematischen Wissenschaften, vol. 316. Springer, Berlin (1998)

    MATH  Google Scholar 

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Acknowledgements

This paper was supported in part by a grant from Ministerio de Economía y Competitividad (MTM 2013-46374-P), Spain. We would like to thank the referee for his/her careful reading of the manuscript and several useful comments which have improved the presentation of the paper.

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  1. Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Madrid, Spain

    José M. Rodríguez

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  1. José M. Rodríguez
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Correspondence to José M. Rodríguez.

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Rodríguez, J.M. Zeros of Sobolev Orthogonal Polynomials via Muckenhoupt Inequality with Three Measures. Acta Appl Math 142, 9–37 (2016). https://doi.org/10.1007/s10440-015-0012-7

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