Polynomials with bounds and numerical approximation

Abstract

We discuss the generation of polynomials with two bounds -an upper bound and a lower bound-on compact setsin view on numerical approximation and scientific computing. We show that a composition formula based on a weighted 4-squares Euler identity generates all such polynomials in dimension d = 1. It yields a new algebraic or compositional approach to the classical problem related to polynomials with minimal uniform norm. Higher dimensions are discussed by means of the 8-squares Degen identity and tensorization, and the connection with quaternions algebras is made explicit. Various numerical results illustrate the potentialities of this approach and some implementation details are provided.

Correction to: Polynomials with bounds and numerical approximation

“…If B n−2 = D n−2 = 0, then A n−1 = A n−1 and C n−1 = C n−1 which corresponds to the original proof [1]. The remaining part of the proof is the same.…”

“…We use the notations of [1]. The missprint shows up in formula (24), where D was replaced by C in the expressions of b and c. Formula (24) now becomes…”

“…Nevertheless the proof must be corrected due to one missprint and missing low order contributions. This correction has its importance in case one desires to implement the method, which is part of our current research.We use the notations of [1]. The missprint shows up in formula (24), where D was replaced by C in the expressions of b and c. Formula (24) now becomes…”

Algorithms For Positive Polynomial Approximation

“…1.21] for a proof based on complex valued trigonometric polynomials. A recent proof in real algebra is available in [2]. Theorem 1.1 (Lukács).…”

“…• The first solution uses an elementary idea proposed in [2]. Since it has a change of sign, it is called an oscillating polynomial in the core of the paper.…”