Bounds-constrained polynomial approximation using the Bernstein basis

TL;DR

This paper develops a method for approximating smooth functions with polynomials that satisfy bounds constraints on Bernstein coefficients, using an inequality-constrained optimization approach, applicable to multivariate cases.

Contribution

It introduces a novel algorithm for bounds-constrained polynomial approximation using Bernstein basis, including extensions for equality constraints and multivariate polynomials.

Findings

Algorithm effectively finds Bernstein coefficients satisfying bounds.

Method extends to multivariate polynomials over a simplex.

Incorporates equality constraints like mass preservation.

Abstract

A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In this paper, we consider the problem of approximating functions by polynomials whose Bernstein coefficients with respect to a given degree satisfy such bounds, which implies such bounds on the approximant. We frame the problem as an inequality-constrained optimization problem and give an algorithm for finding the Bernstein coefficients of the exact solution. Additionally, our method can be modified slightly to include equality constraints such as mass preservation. It also extends naturally to multivariate polynomials over a simplex.