Sums of entire functions having only real zeros

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 obtai...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2007-12, Vol.135 (12), p.3857-3866
Main Authors: Adams, Steven R., Cardon, David A.
Format: Article
Language:English
Subjects:
Algebra
Entire functions
Exact sciences and technology
Exponential functions
Fourier transformations
Functions of a complex variable
General mathematics
General, history and biography
Mathematical analysis
Mathematical functions
Mathematical theorems
Mathematics
Multiplicity of function roots
Number theory
Polynomials
Real functions
Sciences and techniques of general use
Statistical theories
Zero
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Summary: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.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-07-09103-4