Let $f: \Omega_\rho \to \mathbb{C}$ be holomorphic and suppose that $f$ extends continuously to $\overline{\Omega}_\rho$. Assume that there is a constant $M \ge 0$ such that
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\begin{align*}
|f(z)| \le M
\end{align*}
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for every $z \in \overline{\Omega}_\rho$.
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For $n \in \mathbb{N}_0$, let $T_n: [-1,1] \to \mathbb{R}$ denote the $n$th Chebyshev polynomial, characterized by $T_n(\cos \theta)=\cos(n\theta)$ for every $\theta \in \mathbb{R}$. Define the Chebyshev coefficients of $f|_{[-1,1]}$ by