Sums of entire functions having only real zeros

Details Sums of entire functions having only real zeros Sums of entire functions having only real zeros

2006-08-12

arxiv.org/abs/math/0608297v2

Mathematics

Complex Variables

10 pages, corrected minor typos

Scientific paper

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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