A new formulation of Bernstein-Bezier based smoothness conditions forpp functions

In this note, we establish a new formulation of smoothness conditions for piecewise polynomial (: =pp) functions in terms of the B-net representation in the general n-dimensional setting. It plays an important role for 2-dimensional setting in the constructive proof of the fact that the spaces of polynomial splines with smoothness r and total degree k≥3r+2 over arbitrary triangulations achieve the optimal approximation order with the approximation constant depending only on k and the smallest angle of the partition in 5].