Entire periodic functions with plane zeros

Asymptotics of Zeros for Some Entire Functions

We study the asymptotics of zeros for entire functions of the form sin z+ ∫ 1 −1 f(t)e dt with f belonging to a space X →֒ L1(−1, 1) possessing some minimal regularity properties.

Convolution Operators and Zeros of Entire Functions

Let G(z) be a real entire function of order less than 2 with only real zeros. Then we classify certain distributions functions F such that the convolution (G ∗ dF )(z) = ∫∞ −∞G(z − is) dF (s) has only real zeros.

Conformal Mappings And Entire Functions With Real Zeros

Convolution Operators and Entire Functions with Simple Zeros

Let G(z) be an entire function of order less than 2 that is real for real z with only real zeros. Then we classify certain distribution functions F such that the convolution (G ∗ dF )(z) = ∫∞ −∞ G(z − is) dF (s) of G with the measure dF has only real zeros all of which are simple. This generalizes a method used by Pólya to study the Riemann zeta function.

Differential operators and entire functions with simple real zeros

Let φ and f be functions in the Laguerre–Pólya class. Write φ(z) = e−αzφ1(z) and f (z) = e−βzf1(z), where φ1 and f1 have genus 0 or 1 and α,β 0. If αβ < 1/4 and φ has infinitely many zeros, then φ(D)f (z) has only simple real zeros, where D denotes differentiation.  2004 Elsevier Inc. All rights reserved.