Multivariate F-splines and Fractional Box Splines

We introduce multivariate F-splines, including multivariate F-truncated powers T _f (?|M) and F-box splines B _f (?|M). The classical multivariate polynomial splines and multivariate E-splines can be considered as a special case of multivariate F-splines. We document the main properties of T _f (?|M) and B _f (?|M). Using T _f (?|M), we extend fractional B-splines to fractional box splines and show that these functions satisfy most of the properties of the traditional box splines. Our work unifies and generalizes results due to Dahmen-Micchelli, de Boor-Höllig, Ron and Unser-Blu, and also presents a new tool for computing the integration over polytopes.