Minimizing the quasi-interpolation error for bivariate discrete quasi-interpolants

We define a class of discrete quasi-interpolants based on bivariate box splines by imposing the exactness on a space of polynomials of total degree, depending on the box spline and minimizing a constant appearing in the leading term of an appropriate quasi-interpolation error estimate. We give some C1 quadratic and C2 quartic examples and compare them with other well-known quasi-interpolants.