We consider discrete quasi-interpolants based on C~1 quadratic box-splines on uniform crisscross triangulations of a rectangular domain. The main problem consists in finding good (if not best) coefficient functionals, associated with boundary box-splines, giving both an optimal approximation order and a small infinity norm of the operator. Moreover, we want that these functionals only involve data points inside the domain. They are obtained either by minimizing their infinity norm w.r.t. a finite number of free parameters, or by inducing superconvergence of the operator at some specific points lying near or on the boundary.
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