Abstract:
Let $a_k<b_k<a_{k+1},\ k\in\mathbb{Z},\ I_k=(a_k, b_k),\ J_k=[b_k, a_{k+1}].$ We assume that $|I_k|\asymp |J_k|,\ a_k \xrightarrow [k \to +\infty] \ \infty,\ a_k \xrightarrow [k \to -\infty] \ -\infty$ and $|J_k|\asymp \frac{1}{|a_k|^\alpha},\ |k|\rightarrow\infty$, $\alpha>0$. The distribution of $\{J_k\}$ satisfies some regularity conditions, $ E=\bigcup\limits_{k\in \mathbb{Z}} J_k$. A bounded function $f$ is defined on $E$ and satisfies the $s$-Holder condition, $0<s<1$. We put $l_k=\frac{1}{2}|J_k|$, $\xi_k =\frac{1}{2}(b_k+a_{k+1})$, $ k\in\mathbb{Z}$. The function $\rho_t(x)$ is defined for $x\in J_k$ and $0<t\leqslant 1,\ k\in\mathbb{Z}$, as follows: \begin{equation}\notag \rho_t(x)=\begin{cases} ( \sqrt{l_k^2-(x-\xi_k)^2}+t)\cdot\dfrac{t}{|I_k|}, 0<t<\frac{1}{2}, t, \frac{1}{2}\leq t\leqslant 1. \end{cases} \end{equation} We prove the following claim.
Theorem.
There is a constant$c_f$such that an entire function$F_{\sigma}$can be found for any$\sigma\geq 1$with \begin{equation}\notag |F_{\sigma}(z)|\leq c_{F_\sigma}e^{2\sigma|\Im z|},\ z\in \mathbb{C}, \end{equation}
\begin{equation}\notag |f(x)-F_{\sigma}(x)|\leq c_f\rho^s_{\frac{1}{\sigma}}(x),\ x\in E. \end{equation}
Key words and phrases:entire functions of exponential type, approximation, Hölder classes.
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