TY - JOUR
T1 - Entire functions of exponential type represented by pseudo-random and random Taylor series
AU - Borichev, Alexander
AU - Nishry, Alon
AU - Sodin, Mikhail
N1 - Publisher Copyright:
© 2017, Hebrew University Magnes Press.
PY - 2017/10/1
Y1 - 2017/10/1
N2 - We study the influence of the multipliers ξ(n) on the angular distribution of zeroes of the Taylor series Fξ(z)=∑nn)znn!. We show that the distribution of zeroes of Fξ is governed by certain autocorrelations of the sequence ξ. Using this guiding principle, we consider several examples of random and pseudo-random sequences ξ and, in particular, answer some questions posed by Chen and Littlewood in 1967. As a by-product, we show that if ξ is a stationary random integer-valued sequence, then either it is periodic, or its spectral measure has no gaps in its support. The same conclusion is true if ξ is a complex-valued stationary ergodic sequence that takes values in a uniformly discrete set.
AB - We study the influence of the multipliers ξ(n) on the angular distribution of zeroes of the Taylor series Fξ(z)=∑nn)znn!. We show that the distribution of zeroes of Fξ is governed by certain autocorrelations of the sequence ξ. Using this guiding principle, we consider several examples of random and pseudo-random sequences ξ and, in particular, answer some questions posed by Chen and Littlewood in 1967. As a by-product, we show that if ξ is a stationary random integer-valued sequence, then either it is periodic, or its spectral measure has no gaps in its support. The same conclusion is true if ξ is a complex-valued stationary ergodic sequence that takes values in a uniformly discrete set.
UR - http://www.scopus.com/inward/record.url?scp=85037036104&partnerID=8YFLogxK
U2 - 10.1007/s11854-017-0037-0
DO - 10.1007/s11854-017-0037-0
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AN - SCOPUS:85037036104
SN - 0021-7670
VL - 133
SP - 361
EP - 396
JO - Journal d'Analyse Mathematique
JF - Journal d'Analyse Mathematique
IS - 1
ER -