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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 $x \in [-1,1]$.

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  • $\begingroup$ @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? $\endgroup$ – Television Oct 13 '18 at 18:49

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