  We study the approximation capabilities of two families of univariate
polynomials that arise in applications of quantum signal processing. Although
approximation only in the domain $[0,1]$ is physically desired, these
polynomial families are defined by bound constraints not just in $[0,1]$, but
also with additional bound constraints outside $[0,1]$. One might wonder then
if these additional constraints inhibit their approximation properties within
$[0,1]$. The main result of this paper is that this is not the case -- the
additional constraints do not hinder the ability of these polynomial families
to approximate arbitrarily well any continuous function $f:[0,1] \rightarrow
[0,1]$ in the supremum norm, provided $f$ also matches any polynomial in the
family at $0$ and $1$. We additionally study the specific problem of
approximating the step function on $[0,1]$ (with the step from $0$ to $1$
occurring at $x=\frac{1}{2}$) using one of these families, and propose two
subfamilies of monotone and non-monotone approximations. For the non-monotone
case, under some additional assumptions, we provide an iterative heuristic
algorithm that finds the optimal polynomial approximation.
