2021/11/13 by Sarkar, Rahul, Yoder, Theodore J.
#41A10 #41A29 (Primary) #65D15 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.2111.07182
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] → [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=(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.