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.
Best Answer
You are looking for the well-known Remez algorithm, which dates to 1934. For a practical implementation (with an excellent description), I highly recommend the Chebfun function
remez()and this paper that describes it.