# From polynomials to Chebyshev polynomials

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 $$0. The problem of finding the “optimal” values for the $$\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.