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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Sums of entire functions having only real zeros


Authors: Steven R. Adams and David A. Cardon
Journal: Proc. Amer. Math. Soc. 135 (2007), 3857-3866
MSC (2000): Primary 30C15; Secondary 30D05
Published electronically: August 29, 2007
MathSciNet review: 2341936
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Abstract: We show that certain sums of products of Hermite-Biehler entire functions have only real zeros, extending results of Cardon. As applications of this theorem, we construct sums of exponential functions having only real zeros, we construct polynomials having zeros only on the unit circle, and we obtain the three-term recurrence relation for an arbitrary family of real orthogonal polynomials. We discuss a similarity of this result with the Lee-Yang Circle Theorem from statistical mechanics. Also, we state several open problems.


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Additional Information

Steven R. Adams
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602

David A. Cardon
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: cardon@math.byu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09103-4
PII: S 0002-9939(07)09103-4
Received by editor(s): August 9, 2006
Published electronically: August 29, 2007
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2007 American Mathematical Society