De Bruijn's question on the zeros of Fourier transforms

Let f (z) be a real entire function of genus 1*, Δ ≥ 0, and suppose that for each ε > 0, all but a finite number of the zeros of f(z) lie in the strip |Im z| ≤ ≤ Δ + ε. Let λ be a positive constant such that lim sup_{r → ∞} log M(r; f)/r² < 1/(4λ). It is shown that for each ε > 0, all but a finite number of the zeros of the entire function e ^-λD2 f(z) := ∑_m=0^∞ (-λ)^mf^(2m)(z)/m! lie in the strip |Im z| ≤ √max{Δ² - 2λ, 0} + ε; and if Δ ² < 2λ, then all but a finite number of the zeros of e ^-λD2 f(z) are real and simple. As a consequence, de Bruijn's question whether the functions e^λt2, λ > 0, are strong universal factors is answered affirmatively.