Why Bernstein Polynomials Are Better: Fuzzy-Inspired Justification

Authors

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

Publication Date

12-2011

Comments

Technical Report: UTEP-CS-11-62

Abstract

It is well known that an arbitrary continuous function on a bounded set -- e.g., on an interval [a,b] -- can be, with any given accuracy, approximated by a polynomial. Usually, polynomials are described as linear combinations of monomials. It turns out that in many computational problems, it is more efficient to represent a polynomial as Bernstein polynomials -- e.g., for functions of one variable, a linear combination of terms (x-a)^k * (b-x)^n-k. In this paper, we provide a simple fuzzy-based explanation of why Bernstein polynomials are often more efficient, and we show how this informal explanation can be transformed into a precise mathematical explanation.