It is known that the best uniform approximation for a real function defined in interval $[1,-1]$ is via the Chebyshev polynomials. ([see optimal polynomials])1. Such polynomials are also called min-max polynomials (MMP), and they are known to be unique.
Now, I want to uniformly approximate an analytic function defined over the unit disk (centred at the origin) in the complex plane, say $e^{z}$.
Do such unique polynomials exist for this case?
I am reading this monogram by N.L Trefthen. But it seems it was open problem at the time of publication (1988).
What is the current status?
