Let $t\_1 < t\_2 < \cdots <t\_m$ be real, and $X = \cup\_{i=1}^{m-1} (t\_i, t\_{i+1})$ be a union of real open intervals. Let $f:X \rightarrow \{-1, 1\}$ be any piecewise constant function of form
$$
f(x) = 
 \begin{cases} 
 a\_1 & \text{ if } t\_1 < x < t\_2 \\
 a\_2 & \text{ if }t\_2 < x < t\_3 \\
 \vdots \\
a\_{m-2} & \text{ if } t\_{m-2} < x < t\_{m-1} \\ 
 a\_{m-1} & \text{ if } t\_{m-1} < x < t\_m 
 \end{cases}
$$
Where $a\_i \in \{-1, 1\}$, and $a\_{i} = -a\_{i+1}$ for $i = 1, ..., m-1$.

I have a number of questions regarding polynomial approximations of such a function $f$:

1. Can we always find a sequence of polynomials $(p\_n)$ so that $(p\_n)$ converge pointwise to $f$, AND we have some fixed (not arbitrary) global error bound, say $1$, such that $|p\_n(x) - f(x)| \leq 1$ for all $x \in X$ and $n \in \mathbb{N}$?
2. If so, are such polynomials easy to find?
3. How quickly do we get convergence?

I am aware that, upon picking a suitable inner product, we can use any collection of orthonormal polynomials to make approximations of functions. For example I know the Chebyshev, Bernstein, Jacobi etc. polynomials can be used to approximate continuous functions on bounded intervals, but I have found no theorem that says we can use these to construct approximations for arbitrary piecewise constant functions like the one given above.

Indeed, it is easy to find a polynomial approximation for the Heaviside Step function for example, however it is unclear how, or if this an be done for more complicated step functions.