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Density theorems with applications in quantum signal processing.
- Source :
-
Journal of Computational & Applied Mathematics . Oct2023, Vol. 430, pN.PAG-N.PAG. 1p. - Publication Year :
- 2023
-
Abstract
- 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 03770427
- Volume :
- 430
- Database :
- Academic Search Index
- Journal :
- Journal of Computational & Applied Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 163549104
- Full Text :
- https://doi.org/10.1016/j.cam.2023.115243