Theoretical Explanation of Bernstein Polynomials' Efficiency: They Are Optimal Combination of Optimal Endpoint-Related Functions

Authors

Jaime Nava, The University of Texas at El PasoFollow
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

7-2011

Comments

Technical Report: UTEP-CS-11-37

Abstract

In many applications of interval computations, it turned out to be beneficial to represent polynomials on a given interval [x_-, x₊] as linear combinations of Bernstein polynomials (x- x _- )^k * (x₊ - x)^n-k. In this paper, we provide a theoretical explanation for this empirical success: namely, we show that under reasonable optimality criteria, Bernstein polynomials can be uniquely determined from the requirement that they are optimal combinations of optimal polynomials corresponding to the interval's endpoints.