Three-directional box-splines: characterization and efficient evaluation

We propose a new characterization of three-directional box-splines, which are well adapted for interpolation and approximation on hexagonal lattices. Inspired by a construction already applied with success for exponential splines and hex-splines, we characterize a box-spline as a convolution of a generating function, which is a Green function of the spline's associated differential operator, and a discrete filter that plays the role of a localization operator. This process leads to an elegant analytical expression of three-directional box-splines. It also brings along a particularly efficient implementation