Algebraic manipulation in the Bernstein form made simple via convolutions

Traditional methods for algebraic manipulation of polynomials in Bernstein form try to obtain an explicit formula for each coefficient of the result of a given procedure, such us multiplication, arbitrarily high degree elevation, composition, or differentiation of rational functions. Whereas this strategy often furnishes involved expressions, these operations become trivial in terms of convolutions between coefficient lists if we employ the scaled Bernstein basis, which does not include binomial coefficients. We also carry over this scheme from the univariate case to multivariate polynomials, Bézier simplexes of any dimension and B-bases of other functional spaces. Examples of applications in geometry processing are provided, such as conversions between the triangular and tensor-product Bézier forms.