Constrained degree reduction of polynomials in Bernstein–Bézier form over simplex domain

In this paper we show that the orthogonal complement of a subspace in the polynomial space of degree n over d  -dimensional simplex domain with respect to the L2L2-inner product and the weighted Euclidean inner product of BB (Bézier–Bernstein) coefficients are equal. Using it we also prove that the best constrained degree reduction of polynomials over the simplex domain in BB form equals the best approximation of weighted Euclidean norm of coefficients of given polynomial in BB form from the coefficients of polynomials of lower degree in BB form.