Hermite Multiplier Sequences and Their Associated Operators


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.

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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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Correspondence to Tamás Forgács.

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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.

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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

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  • Hermite expansions
  • Linear operators
  • Zeros of polynomials

Mathematics Subject Classification

  • 30C15
  • 26C10