Density theorems with applications in quantum signal processing

Understand

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 $1$.