A projection algorithm on the set of polynomials with two bounds

TL;DR

This paper introduces a new nonlinear projection algorithm for approximating functions with polynomials constrained by two bounds, leveraging algebraic structures to ensure stability and provide error estimates.

Contribution

It develops a novel projection method onto polynomials with two bounds using quaternion algebra and Euler four-squares, with theoretical analysis and numerical validation.

Findings

The algorithm is stable and continuous.

Error estimates are provided for the projection.

Numerical tests demonstrate effectiveness.

Abstract

The motivation of this work stems from the numerical approximation of bounded functions by polynomials satisfying the same bounds. The present contribution makes use of the recent algebraic characterization found in [B. Despr\'es, Numer. Algorithms, 76(3), (2017)] and [B. Despr\'es and M. Herda, Numer. Algorithms, 77(1), (2018)] where an interpretation of monovariate polynomials with two bounds is provided in terms of a quaternion algebra and the Euler four-squares formulas. Thanks to this structure, we generate a new nonlinear projection algorithm onto the set of polynomials with two bounds. The numerical analysis of the method provides theoretical error estimates showing stability and continuity of the projection. Some numerical tests illustrate this novel algorithm for constrained polynomial approximation.