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A Characterization of Multiplier Sequences for Generalized Laguerre Bases

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Abstract

We give a complete characterization of multiplier sequences for generalized Laguerre bases. We also apply our methods to give a short proof of the characterization of Hermite multiplier sequences achieved by Piotrowski.

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References

  1. Blakeman, K., Davis, E., Forgács, T., Urabe, K.: On Legendre multiplier sequences. Mo. J. Math. Sci. 24, 7–23 (2012)

    Google Scholar 

  2. Bleecker, D., Csordas, G.: Hermite expansions and the distribution of zeros of entire functions. Acta Sci. Math. 67, 177–196 (2001)

    MATH  MathSciNet  Google Scholar 

  3. Borcea, J., Brändén, P.: Pólya–Schur master theorems for circular domains and their boundaries. Ann. Math. 170(1), 465–492 (2009)

    Article  MATH  Google Scholar 

  4. Borcea, J., Brändén, P.: Multivariate Pólya–Schur classification problems in the Weyl algebra. Proc. Lond. Math. Soc. 101, 73–104 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Borcea, J., Brändén, P.: The Lee–Yang and Pólya–Schur programs. I. Linear operators preserving stability. Invent. Math. 177, 541–569 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Brändén, P.: The Lee–Yang and Pólya–Schur programs. III. Zero-preservers on Bargmann-Fock spaces. Am. J. Math. (to appear). arXiv:1107.1809

  7. Craven, T., Csordas, G.: Composition theorems, multiplier sequences and complex zeros decreasing sequences. In: Barsegian, G., Laine, I., Yang, C.C. (eds.) Value Distribution Theory and Related Topics. Advances in Complex Analysis and Its Applications, vol. 3, pp. 131–166. Kluwer Academic, Dordrecht (2004)

    Chapter  Google Scholar 

  8. Craven, T., Csordas, G.: The Gauss-Lucas theorem and Jensen polynomials. Trans. Am. Math. Soc. 278, 415–429 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  9. Forgács, T., Piotrowski, A.: Multiplier sequences for generalized Laguerre bases. Rocky Mt. J. Math. (to appear). arXiv:1002.0759 [math.CV]

  10. Laguerre, E.: Oeuvres, vol. I. Gauther-Villars, Paris (1898)

    Google Scholar 

  11. Levin, B.: Distribution of Zeros of Entire Functions. Transl. Math. Monogr., vol. 5. Am. Math. Soc., Providence (1980). xii+523 pp. Translated from Russian by R.P. Boas, J.M. Danskin, F.M. Goodspeed, J. Korevaar, A.L. Shields and H.P. Thielman

    Google Scholar 

  12. Piotrowski, A.: Linear operators and the distributions of zeros of entire functions. Dissertation, University of Hawaii at Manoa (2007)

  13. Pólya, G., Schur, J.: Über zwei arten von faktorenfolgen in der theorie der algebraischen gleichungen. J. Reine Angew. Math. 144, 89–113 (1914)

    MATH  Google Scholar 

  14. Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials. London Math. Soc. Monogr. (N. S.), vol. 26. Oxford University Press, New York (2002)

    MATH  Google Scholar 

  15. Rainville, E.D.: Special Functions. Macmillan, New York (1960)

    MATH  Google Scholar 

  16. Turán, P.: Sur l’algèbre fonctionelle. Compt. Rend. du prem. Congr. des Math. Hongr. 27, 279–290 (1952)

    Google Scholar 

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Acknowledgements

The first author is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation. The research is also supported by the Göran Gustafsson Foundation.

We thank the two anonymous referees for several valuable suggestions that improved the exposition.

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Authors and Affiliations

  1. Department of Mathematics, Royal Institute of Technology, Stockholm, 100 44, Sweden

    Petter Brändén

  2. Department of Mathematics, Stockholm University, Stockholm, 10691, Sweden

    Elin Ottergren

Authors
  1. Petter Brändén
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  2. Elin Ottergren
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Correspondence to Petter Brändén.

Additional information

Communicated by Allan Pinkus.

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Brändén, P., Ottergren, E. A Characterization of Multiplier Sequences for Generalized Laguerre Bases. Constr Approx 39, 585–596 (2014). https://doi.org/10.1007/s00365-013-9204-4

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  • DOI: https://doi.org/10.1007/s00365-013-9204-4

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