Local Quartic C^2 Spline Quasi-interpolation on 3D Bounded Domains

Given a 3D bounded domain, in this paper we present new quasi-interpolating spline schemes, based on trivariate C^2 quartic box splines on type-6 tetrahedral partitions with approximation order four. They are of near-best type, i.e. with coefficient functionals obtained by minimizing an upper bound for their infinity norm. Such quasiinterpolants can be used for the reconstruction of gridded volume data and their higher smoothness is useful, for example, when functions have to be reconstructed with C^2 smoothness. Moreover, we give norm and error bounds. Finally, some numerical tests, illustrating the approximation properties of the proposed quasiinterpolants, and comparisons with other known spline methods are presented.