Institution: | Fakultät für Mathematik, Universität Bielefeld, 4800, Bielefeld 1, Germany Fed. Rep. IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A. |
Abstract: | Let Δ denote the triangulation of the plane obtained by multi-integer translates of the four lines x=0, y=0, x=y and x=?y. By we mean the space of all piecewise polynomials of degree ?k with respect to the scaled triangulation hΔ having continuous partial derivatives of order . We show that the approximation properties of are completely governed by those of the space spanned by the translates of all so called box splines contained in . Combining this fact with Fourier analysis techniques allows us to determine the optimal controlled approximation rates for the above subspace of box splines where μ is the largest degree of smoothness for which these spaces are dense as h tends to zero. Furthermore, we study the question of local linear dependence of the translates of the box splines for the above criss-cross triangulations. |