**In short:** For a given smooth or continuous function, how can we obtain the best $L^{\infty }$ approximating polynomial?


Jackson (1911) proved that there is a best approximating polynomial in the $L^{\infty}$ sense. The proof Can be found in the references below. The theorem is


Let $I=\lbrack -1,1]$, then there is a constant $C>0$ such that for all $f\in C\left( \bar{I} \right)$ we have:


$\| f - \Psi\_{\infty, n} (f) \|\_{\infty } \leq C sup\_{\left| x-y\right| < \frac{2}{n}} \left| f(x) - f(y) \right| $


Where $\Psi\_{\infty, n} (f) $ is the best interpolating polynomial in the aforementioned sense. However, I couldn't find anything about what would be this polynomial, or how to build it. I would assume that some advance was made since, but I couldn't find it in these textbooks.


Thank you


*References*


[1] Funaro, Polynomial Approximation of Differential Equations, theorem 6.1.2


[2] Davis, Approximation Theory, theorem 13.3.7.



$L^{\infty}$ polynomial approximation
You are looking for the well-known [Remez algorithm](https://en.wikipedia.org/wiki/Remez_algorithm), which dates to 1934. For a practical implementation (with an excellent description), I highly recommend the [Chebfun function `remez()`](http://www.chebfun.org/examples/approx/BestApprox.html) and [this paper that describes it](http://dx.doi.org/10.1007/s10543-009-0240-1).


