Bernstein–Bézier representation and near-minimally normed discrete quasi-interpolation operators

We are concerned with the construction of bivariate box-spline discrete quasi-interpolants with small infinity norms and optimal approximation orders. They are defined by minimizing a sharp upper bound of the uniform norm which is derived from the Bernstein–Bézier representation of the corresponding fundamental function. We detail the construction of such quadratic and quartic quasi-interpolants.