A closed PP form of box splines via Green's function decomposition
A set construction scheme is presented for the class of non-degenerate box splines. This scheme reparably decomposes the Green's function of a box spline, yielding its explicit piecewise polynomial form. It is possible to use the well-known recursive formulation to obtain these polynomial pieces, but that procedure is quite expensive. The fast evaluation schemes using piecewise polynomial form of box splines are created. The spatial Fourier form of Green's function and the spatial form of the difference operator are used. It is proved that, under certain conditions, the proposed decomposition procedure is of asymptotically orders of magnitude lower than the recursive procedure. This allows to evaluate box splines with more direction vectors than what would be feasible under the recursive scheme. The exemplary constructions for some known box splines are provided.
piecewise polynomial form of box splines
scientific article; zbMATH DE number 6881221