I was wondering how they got from the polynomial to a Chebyshev polynomial as outlined here:

In order to obtain “strong” stability, we replace the condition ($2.7$) by $$\left|\prod_{j=1}^N (1-\tau_j\lambda)\right| \leq K \qquad \forall \ \lambda \in [\mu,\lambda_\max], \tag{2.8}$$ where $\mu$ is some number in the interval $(0, \lambda_\min]$, and $K$ is some number $0<K<1$. The problem of finding the “optimal” values for the $\tau_j$’s can then be reformulated as

Find $\tau_1$, $\tau_2$, $\ldots$, $\tau_N$ such that $p_N(\lambda) =\prod_{j=1}^N (1-\tau_j\lambda)$ satisfies \begin{array}{l@{\qquad}l} |p_N(\lambda)| \leq K \quad \forall \ \lambda \in [\mu, \lambda_\max] && \text{(STABILITY),} \\ \displaystyle |p'_N(0)| =\sum_{j=1}^N \tau_j \quad \text{maximal} && \text{(OPTIMALITY).} \end{array}

Using the remarkable optimality properties of the Chebyshev polynomials $T_N(\cdot)$ of degree $N$, Markoff [$5$] ($1892!$), we have that if $K$ is given by $$K=1/T_N \left(\frac{\lambda_\max +\mu}{\lambda_\max -\mu}\right)$$

Anyone knows?

The actual paper, I'm referring to is here.


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