Sufficient Conditions and Bounding Properties for Control Functions Using Bernstein Expansion

Bernstein expansion of a polynomial function has linear and quadratic rates of convergence to the original function. In this paper, we extend a direct approximation method by the minimum and maximum Bernstein control points to multivariate polynomials and continuous rational functions over boxes. Furthermore, we explore the rate of convergence and properties of Bernstein basis and illustrate the advantages of this approach through its applications for positivity of nonlinear functions. To this end, sharpness, minimization, degree elevation and convergence properties of polynomials are extended to the multivariate rational polynomial Bernstein case. Subsequently, local and global positive values of control Bernstein points are computed. Finally, several valid optimization bounds for the degree of Bernstein basis and the width of a box are given