# What is the rational function that deviates least from $0$?

It is a well known result that among the set of polynomials of degree $$n$$ with leading coefficient $$1$$, the $$n^{th}$$ order Chebyshev polynomial $$T_n (x)$$ minimizes the (point-wise maximum) deviation from $$0$$ over $$[-1,1]$$. Are there any such results on what is the rational function which has minimum maximum deviation from $$0$$ over $$[-1,1]$$ (or some compact)?

i.e. find the rational function $$\frac{P(x)}{Q(x)}$$, where

• for $$x \in [-1,1]$$, $$Q(x) \ne 0$$
• $$P(x), Q(x)$$ have degree $$p,q$$ respectively

which has minimum point-wise maximum deviation from $$0$$ for $$x \in [-1,1]$$ among the class of rational functions having numerator degree $$p$$ and denominator degree $$q$$.

Edit: Also include the constraint $$|P(x)| \le 1$$, $$|Q(x)| \le 1$$ for $$x \in [-1,1]$$.

• @metamorphy I think I will drop the leading co-efficient condition. It will be nice to find a clean closed form expression, but maybe I am asking too much? – Television Oct 13 '18 at 18:49