Theoretical Explanation of Bernstein Polynomials' Efficiency: They Are Optimal Combination of Optimal Endpoint-Related Functions
Jaime Nava, Владик Крейнович · scholarworks - UTEP (The University of Texas at El Paso) · 2011
In many applications of interval computations, it turned out to be beneficial to represent polynomials on a given interval [x; x] as linear com-binations 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 end-points.