Why bernstein polynomials are better: Fuzzy-inspired justification

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 or by a piece-wise polynomial function (spline). Usually, polynomials are described as linear combinations of monomials. It turns out that in many computational problems, it is more efficient to represent each polynomial as a Bernstein polynomial - 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 than linear combinations of monomials, and we show how this informal explanation can be transformed into a precise mathematical explanation.