On the construction of trivariate near-best quasi-interpolants based on C2 quartic splines on type-6 tetrahedral partitions

The construction of new quasi-interpolants (QIs) having optimal approximation order and small infinity norm and based on a trivariate $C^2$ quartic box spline is addressed in this paper. These quasi-interpolants, called near-best QIs, are obtained in order to be exact on the space of cubic polynomials and to minimize an upper bound of their infinity norm which depends on a finite number of free parameters in a tetrahedral sequence defining the coefficients of the QIs. We show that this problem has always a unique solution, which is explicitly given. We also prove that the sequence of the resulting near-best quasi-interpolants converges in the infinity norm to the Schoenberg operator.