# Error upper bound of minimax polynomial approximation over $[a,b] \cup [c,d]$

I found that Jackson's inequality in approximation theory provides us a nice upper bound on the (infinite-norm) error of minimax polynomial over an interval $$[a,b]$$ as noted in [1, p. 16]. Can it be extended to more general cases such as $$[a,b] \cup [c,d]$$?

To be precise, I want to find an error upper bound of minimax approximation of the step function $$\chi_{(0,\infty)}$$ over $$[-1,-\epsilon] \cup [\epsilon, 1]$$ for small $$\epsilon > 0$$.

Thanks for the help in advance.

[1] Pachón, Ricardo, and Lloyd N. Trefethen. "Barycentric-Remez algorithms for best polynomial approximation in the chebfun system." BIT Numerical Mathematics 49.4 (2009): 721.