Minimax Function Approximation Lab

\[\begin{align*}Let\ f\ be\ a\ continuos\ function\ from\ \left[a, b\right]\ to\ \mathrm{R}.\end{align*}\] \[\begin{align*}Among\ all\ the\ polynomials\ of\ degree\le\ n\ the\ polynomial\ p_{n}^{*}\ minimizes\ the\ maximum\ norm\end{align*}\] \[\begin{align*}\ of\ the\ approximation\ error\ \left\|f\left(x\right)-p\left(x\right)\right\|^{\infty}_{\left[a,b\right]}=\max_{x\in \left[a,b\right]}{\left|\left(f\left(x\right)-p\left(x\right)\right)\right|},\end{align*}\] \[\begin{align*}if\ and\ only\ if\ there\ are\ n+2\ points\ a\le x_{0}\le x_{1}\le \cdots \le x_{n}\le x_{n+1}\le b\end{align*}\] \[\begin{align*}such\ that\ f(x_{i})-p^{*}_{n}(x_{i})=\sigma(-1)^{i}\left\| f-p^{*}_{n} \right\|^{\infty }_{[a,b]},\ where\ \sigma\ is\ fixed\ as\ either\ +1\ or\ -1.\end{align*}\] \[\begin{align*}In\ other\ words,the\ error\ function\ f-p^{*}_{n}\ \end{align*}\] \[\begin{align*}reaches\ its\ absolute\ maximum\ (at\ least)\ n+2\ times\ with\ alternating\ sign.\end{align*}\]