2020/01/15 by Igor Chyzhykov, Chyzhykov, Igor, Jianren Long +1
Computer Science · Mathematics · #30C15 #30H99 #30J99 #34C10 #Algebraic and Geometric Analysis #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2001.06378
openalex publication_date 2020/01/15 · openalex created_date 2020/01/23 · openalex updated_date 2026/07/28
Let (zk) be a sequence of distinct points in the unit disc \mathbbD without limit points there. We are looking for a function a(z) analytic in \mathbbD and such that possesses a solution having zeros precisely at the points zk, and the resulting function a(z) has `minimal' growth. We focus on the case of non-separated sequences (zk) in terms of the pseudohyperbolic distance when the coefficient a(z) is of zero order, but sup_z∈ \mathbbD (1-|z|)p |a(z)|=+∞ for any p>0. We established a new estimate for the maximum modulus of a(z) in terms of the functions nz(t)=∑|zk-z|≤ t 1 and Nz(r)=∫0r ((nz(t)-1)+)/(t)dt. The estimate is sharp in some sense. The main result relies on a new interpolation theorem.