2020/05/06 by Б. Н. Хабибуллин, Khabibullin, B. N., Анна Евгениевна Салимова +1
Engineering · Mathematics · #30D15 (Primary) 31A05 #30D20 #31A15 (Secondary) #Complex Variables (math.CV) #FOS: Mathematics #Material Science and Thermodynamics #Meromorphic and Entire Functions #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2005.05781
openalex publication_date 2020/05/06 · openalex created_date 2020/05/21 · openalex updated_date 2026/07/28
Previously, we developed the technique of balayage of measures or charges and (δ-)subharmonic functions of finite order onto an closed system of rays S with a vertex at zero on the complex plane \mathbb C. In this article, we use only two kinds of balayage of measure and charge, as well as of subharmonic functions of finite type under the order 1 and their differences. First, it is a classical balayage of the genus q=0 on a system of four closed rays: positive and negative, and real and imaginary semi-axis \mathbb R+, -\mathbb R+, i\mathbb R+, -i\mathbb R. Second, it is two-sided balayage of genus q=1 from the open right and left half-planes \mathbb C\rm rh and \mathbb C\rm lh onto the imaginary axis i\mathbb R. The classical Malliavin-Rubel theorem gives necessary and sufficient conditions of the existence of an entire function of exponential type (we write e.f.e.t.) f\not≡ 0, vanishing on the given positive sequence \sf Z=\\sf zk\k∈ \mathbb N⊂ \mathbb R+ and satisfying the constraint |f|≤ |g| on i\mathbb R, where g is an e.f.e.t., vanishing on positive sequence \sf W=\\sf wk\k∈ \mathbb N⊂ \mathbb R+. A combination of these special balayage processes of genus q=0 and q=1 allows us to extend the Malliavin-Rubel theorem to arbitrary complex sequences \sf Z=\\sf zk\k∈ \mathbb N⊂ \mathbb C separated by a pair of vertical angles from the imaginary axis i\mathbb R, with much more general restrictions ln |f|≤ M on the imaginary axis i\mathbb R, where M is an subharmonic function of finite type under the order 1.