On the optimal approximation rates for criss-cross finite element spaces

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 lk, hμ we mean the space of all piecewise polynomials of degree ≤k with respect to the scaled triangulation hΔ having continuous partial derivatives of order ≤μ on R2. We show that the approximation properties of lk, hμ are completely governed by those of the space spanned by the translates of all so called box splines contained in lk,hμ. 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. © 1984.