Let $(z_k)$ be a sequence of distinct points in the unit disc $\mathbb {D}$ without limit points there. We are looking for a function $a(z)$ analytic in $\mathbb {D}$ and such that possesses a solution having zeros precisely at the points $z_k$, and the resulting function $a(z)$ has ‘minimal’ growth. We focus on the case of non-separated sequences $(z_k)$ in terms of the pseudohyperbolic distance when the coefficient $a(z)$ is of zero order, but $\sup _{z\in {\mathbb D}}(1-|z|)^p|a(z)| = + \infty$ for any $p > 0$. We established a new estimate for the maximum modulus of $a(z)$ in terms of the functions $n_z(t)=\sum \nolimits _{|z_k-z|\le t} 1$ and $N_z(r) = \int_0^r {{(n_z(t)-1)}^ + } /t{\rm d}t.$ The estimate is sharp in some sense. The main result relies on a new interpolation theorem.

中文翻译：

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关于线性微分方程解的非分离零序列

让$(z_k)$是单位圆盘中一系列不同的点$\mathbb {D}$那里没有限制点。我们正在寻找一个函数$a(z)$分析在$\mathbb {D}$并且具有在点处精确为零的解$z_k$, 和结果函数$a(z)$有“最小”的增长。我们专注于非分离序列的情况$(z_k)$就伪双曲线距离而言，当系数$a(z)$是零阶的，但是$\sup _{z\in {\mathbb D}}(1-|z|)^p|a(z)| = + \infty$对于任何$p > 0$. 我们建立了一个新的估计最大模量$a(z)$在功能方面$n_z(t)=\sum \nolimits _{|z_k-z|\let} 1$和$N_z(r) = \int_0^r {{(n_z(t)-1)}^ + } /t{\rm d}t.$从某种意义上说，这个估计是尖锐的。主要结果依赖于一个新的插值定理。