Hermite Multiplier Sequences and Their Associated Operators

Abstract

We provide an explicit formula for the coefficient polynomials of a Hermite diagonal differential operator. The analysis of the zeros of these coefficient polynomials yields the characterization of generalized Hermite multiplier sequences which arise as Taylor coefficients of real entire functions with finitely many zeros. We extend our result to functions in \({\mathcal {L}}-{\mathcal {P}}\) with infinitely many zeros, under additional hypotheses.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

References

  1. 1.

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

    MATH  MathSciNet  Google Scholar 

  2. 2.

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

    MATH  Article  Google Scholar 

  3. 3.

    Brändén, P., Ottergren, E.: A characterization of multiplier sequences for generalized Laguerre bases. Constr. Approx. 39, 585–596 (2014)

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Carlitz, L.: A note on Hermite polynomials. Am. Math. Mon. 62, 646–647 (1955)

    MATH  Article  Google Scholar 

  5. 5.

    Chasse, M.: Linear preservers and entire functions with restricted zero loci, Ph.D. dissertation, University of Hawaii (2011). http://scholarspace.manoa.hawaii.edu/bitstream/handle/10125/25933/PhD_2011_Chasse_r.pdf?sequence=2

  6. 6.

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

    MATH  MathSciNet  Article  Google Scholar 

  7. 7.

    Craven, T., Csordas, G.: Jensen polynomials and the Turán and Laguerre inequalities. Pac. J. Math. 136, 241–260 (1989)

    MATH  MathSciNet  Article  Google Scholar 

  8. 8.

    Craven, T., Csordas, G.: Composition theorems, multiplier sequences, and complex zero 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. Kluwer Press, Dordrecht (2004)

    Google Scholar 

  9. 9.

    Csordas, G., Varga, R., Vincze, I.: Jensen polynomials with applications to the Riemann \(\xi \)-function. J. Math. Anal. Appl. 153, 112–135 (1990)

    MATH  MathSciNet  Article  Google Scholar 

  10. 10.

    Dimitrov, D.K., Ben Cheikh, Y.: Laguerre polynomials as Jensen polynomials of Laguerre-Pólya entire functions. J. Comput. Appl. Math. 233, 703–707 (2009)

    MATH  MathSciNet  Article  Google Scholar 

  11. 11.

    Forgács, T., Piotrowski, A.: Multiplier sequences for generalized Laguerre bases. Rocky Mt. J. Math. 43, 1141–1159 (2013)

    MATH  Article  Google Scholar 

  12. 12.

    Iserles, A., Nørsett, S.P.: Zeros of transformed polynomials. SIAM J. Math. Anal. 21(2), 483–509 (1990)

    MATH  MathSciNet  Article  Google Scholar 

  13. 13.

    Iserles, A., Saff, E.B.: Zeros of expansions in orthogonal polynomials. Math. Proc. Camb. Philos. Soc. 105, 559–573 (1989)

    MATH  MathSciNet  Article  Google Scholar 

  14. 14.

    Krantz, S., Rosen, K.H., Zwillinger, D. (eds.): Standard Mathematical Tables and Formulae. CRC Press Inc, Boca Raton (1996)

  15. 15.

    Levin, B. Ja.: Distribution of zeros of entire functions. Trans. Math. Mono. (5), Am. Math. Soc., Providence, RI (1964). Revised ed. (1980)

  16. 16.

    Peetre, J.: Une caractérisation abstraite des opérateurs differentiels. Math. Scand, 7, 211–218 (1959), Erratum, ibid. 8, 116–120 (1960)

  17. 17.

    Piotrowski, A.: Linear operators and the distribution of zeros of entire functions, Ph.D. Dissertation, University of Hawaii (2007). http://scholarspace.manoa.hawaii.edu/bitstream/handle/10125/25932/PhD_2007_Piotrowski_r.pdf?sequence=1

  18. 18.

    Pólya, G., Schur, J.: Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math. 144, 89–113 (1914)

    MATH  MathSciNet  Google Scholar 

  19. 19.

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

    MATH  Google Scholar 

  20. 20.

    Riordan, J.: Combinatorial Identities. Wiley, New York (1968)

    MATH  Google Scholar 

  21. 21.

    Szegő, G.: Orthogonal polynomials. Am. Math. Soc. Colloquium Publications XXIII (1939)

  22. 22.

    Turán, P.: Sur l’algèbre fonctionelle. Comput. Rend. du prem. (Août-2 Septembre 1950) Congr. des Math. Hongr. 27, 279–290 (1952)

  23. 23.

    Turán, P.: Hermite-expansion and strips for zeros of polynomials. Arch. Math. 5, 148–152 (1954)

    MATH  MathSciNet  Article  Google Scholar 

Download references

Acknowledgments

We would like to thank Professor George Csordas for many stimulating discussions, for some illuminating examples, and for his enduring encouragement during the completion of this work. We also express our gratitude to the participants of the University of Hawaii 2013 fall graduate complex analysis seminar, and in particular to Robert Bates for bringing the expression in Eq. (10) to our attention. Finally, we would like to thank the two anonymous referees for their helpful comments on improving the exposition.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Tamás Forgács.

Additional information

Parts of the work were completed while the first author was on sabbatical leave at the University of Hawaii at Manoa, whose support he gratefully acknowledges. The authors would also like to thank their respective institutions for providing financial support for their research.

Communicated by Edward B. Saff.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Forgács, T., Piotrowski, A. Hermite Multiplier Sequences and Their Associated Operators. Constr Approx 42, 459–479 (2015). https://doi.org/10.1007/s00365-015-9277-3

Download citation

Keywords

  • Hermite expansions
  • Linear operators
  • Zeros of polynomials

Mathematics Subject Classification

  • 30C15
  • 26C10