On non-separated zero sequences of solutions of a linear differential equation
| Autor: | Jianren Long, Igor Chyzhykov |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Physics
Zero order Sequence Mathematics - Complex Variables Mathematics::Number Theory General Mathematics 010102 general mathematics Zero (complex analysis) 010103 numerical & computational mathematics Function (mathematics) 01 natural sciences Combinatorics Linear differential equation 34C10 30C15 30H99 30J99 Limit point FOS: Mathematics Complex Variables (math.CV) 0101 mathematics Unit (ring theory) |
| Zdroj: | Proceedings of the Edinburgh Mathematical Society. 64:247-261 |
| ISSN: | 1464-3839 0013-0915 |
| Popis: | 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_{|z_k-z|\le t} 1 $ and $N_z(r)=\int_0^r \frac{(n_z(t)-1)^+}{t}dt.$ The estimate is sharp in some sense. The main result relies on a new interpolation theorem. 17 pages. arXiv admin note: text overlap with arXiv:1401.0797 |
| Databáze: | OpenAIRE |
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