Generalized Popoviciu expansions for Bernstein polynomials of a rational module

We prove that Bernstein polynomials for simple nonsmooth functions such as a rational module can be represented as special sums of a regular structure called “generalized Popoviciu decompositions.” To write generalized expansions, a certain formalism based on combinatorial calculations is developed. Based on the formulas obtained, we obtain a complete description of the convergence set of Bernstein polynomials of a rational module. The connection between the Popoviciu expansions and the distribution of zeros of Bernstein polynomials on the complex plane is discussed. In conclusion, a number of additional, new relations for Bernstein polynomials of a rational module are presented.