Density theorems with applications in quantum signal processing

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.