I'm looking for a resource to find bounds for Chebyshev polynomial approximations of a given degree for the Heaviside function $$ H(x) = \begin{cases} 1 & x>0,\\ 0 & \text{else.} \end{cases} $$
I've done some Googling, but I can't seem to find a good resource on this. Although at the moment, my focus is on getting bounds to approximating this function with polynomials, I know I will wish to study this for other functions as well. Is there a good place to look in general to find the order of convergence of the Chebyshev polynomials for common functions?
Put more concretely, if I have a linear combination of $k$ Chebyshev polynomials \begin{equation} f(x) = \sum_{l=0}^{k-1} c_lT_l(x) \end{equation} what can I say about \begin{equation} \epsilon = \int_{-1}^1 |f(x) - H(x)|dx? \end{equation} Is $\epsilon = $ $O(1/d)$, or $O(1/\text{poly(d)})$, or $O(1/2^d)$?