TY - JOUR
T1 - De Bruijn's question on the zeros of Fourier transforms
AU - Ki, Haseo
AU - Kim, Young One
N1 - Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.
PY - 2003
Y1 - 2003
N2 - 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 supr → ∞ 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 eλt2, λ > 0, are strong universal factors is answered affirmatively.
AB - 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 supr → ∞ 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 eλt2, λ > 0, are strong universal factors is answered affirmatively.
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U2 - 10.1007/BF02788795
DO - 10.1007/BF02788795
M3 - Article
AN - SCOPUS:1842486568
VL - 91
SP - 369
EP - 387
JO - Journal d'Analyse Mathematique
JF - Journal d'Analyse Mathematique
SN - 0021-7670
ER -