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Constructive Approximation

, Volume 42, Issue 3, pp 459–479 | Cite as

Hermite Multiplier Sequences and Their Associated Operators

  • Tamás ForgácsEmail author
  • Andrzej Piotrowski
Article

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.

Keywords

Hermite expansions Linear operators Zeros of polynomials 

Mathematics Subject Classification

30C15 26C10 

Notes

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.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.California State University, FresnoFresnoUSA
  2. 2.University of Alaska SoutheastJuneauUSA

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