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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.

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