Approximation theory, which aims to provide the optimal polynomial function approximating the target function in a given domain such as $x\in[-1,1]$, has been well-developed for real variables. In particular, the Chebyshev polynomial plays an important role in that it often provides near-optimal performance.

My questions are: I) Are there studies/theories for approximation theory for complex variables (i.e., holomorphic functions, especially for $|x|<1$) and maybe some reference on this; II) What is the counterpart of Chebyshev polynomials in this case?

PS: I have also noticed other questions in the channel. Yet they do not address my problems directly.