Inverse theorem of approximation by entire functions of exponential type

O. V. Silvanovich ; N. A. Shirokov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 52, Tome 537 (2024), pp. 104-115

Résumé

Let $I_k=[a_k,b_k],\ J_k=[b_k,a_{k+1}],\ b_k$, be segments on the real axis tending to $+\infty$ and to $-\infty$. Assume that these segments satisfy the conditions $|I_k|=2^{-n\alpha}$ if $I_k\subset [2^{n},2^{n+1}] $ or $I_k\subset [-2^{n+1},-2^{n}] $, $\alpha>0$ being fixed, $n\geq n_0$. Assume also that there exists a constant $c_1>0$ such that $ 2^{n_0}\cdot 2^{-n\alpha}\leq|J_k|\leq c_1\cdot2^{n_0}\cdot 2^{-n\alpha}$ if $J_k\subset [2^{n},2^{n+1}]$ or $J_k\subset [-2^{n+1},-2^{n}],k\in\mathbb{Z}$. Put $E=\bigcup\limits_{k\in\mathbb{Z}}J_k$. Denote by $f_{E,1}(z)$ a function subharmonic on $\mathbb{C}$ and satisfying the conditions $f_{E,1}(x)=0,x\in E, f_{E,1}(z)$ is harmonic on $\mathbb{C}\setminus E$, $ \underset{z\rightarrow\infty}{\varlimsup} \dfrac{f_{E,1}(z)}{|z|}=1$, and $g(z)\leq f_{E,1}(z),\ z\in\mathbb{C}$, for every function $g$ subharmonic on $\mathbb{C}$ and such that $g(x)\leq 0$, $x\in E$, and $ \underset{z\rightarrow\infty}{\varlimsup}\dfrac{g(z)}{|z|}\leq1.$ We define sets $L_t(E)$ as follows: $$L_t=\{z\in\mathbb{C}: f_{E,1}(z)=t\}$$ and put $\rho_t(x)=\text{dist}(x,L_t(E)),\ x\in E$. Let $T_{\sigma}$ be the set of entire functions of exponential type satisfying the condition $|F_{\sigma}(z)|\leq c_{F_{\sigma}}\exp(\sigma|\text{Im}z|),z\in\mathbb{C}, F_{\sigma}\in T_{\sigma}.$ Denote by $\Lambda^s(E)$ the $s$-Hölder class on $E,\ 0$ of functions bounded on the set $E$. We prove the following result. Theorem 1. Assume that for a function $f$ defined on $ E$ there exist functions $F_{\sigma}\in T_{\sigma}$ such that \begin{equation}{\notag} |f(x)-F_{\sigma}(x)|\leq c_f\rho^s_{\frac{1}{\sigma}}(x),\ x\in E,\ \sigma \geq 1. \end{equation} Then $f \in \Lambda^s(E)$.