Flexible Functional Forms Bernstein Polynomials

Motivated by the economic theory of cost functions, bivariate Bernstein polynomials are considered for approximating shape-restricted functions that are continuous, non-negative, monotone non-decreasing, concave, and homogeneous of degree one. We show the explicit rates of convergence of our approximating polynomials for general functions. We prove some interesting properties of bivariate Bernstein polynomials, including bimonotonicity for concave functions. Moreover, using the classical results, global approximations for shape-restricted functions can be achieved. We also note that concavity violation by the bivariate Bernstein polynomials occurs when the underlying true function ishomogeneous of degree one. However, this violation diminishes as indicces get large.