The Bernstein Expansion for Rational Differentiable Functions in Newton Divided Differences Form

In this paper, we address the Bernstein expansion of rational Differentiable functions in Newton Divided Differences form. We extend the approach of Bernstein to rational polynomials and approximate the truncation error. Additionally, we show essential properties for the approach in the Newton Divided Differences. Subsequently, we prove that rational continuous functions can be optimized by the minimum and maximum Bernstein control points that occur at the corresponding nods.