Bivariate box splines and smooth pp functions on a three direction mesh

Abstract Let S denote the space of bivariate piecewise polynomial functions of degree ⩽ k and smoothness ρ on the regular mesh generated by the three directions (1, 0), (0, 1), (1, 1). We construct a basis for S in terms of box splines and truncated powers. This allows us to determine the polynomials which are locally contained in S and to give upper and lower bounds for the degree of approximation. For ρ = ⌊ (2k − 2) 3 ⌋ , k ≢ 2 (3), the case of minimal degree k for given smoothness ρ, we identify the elements of minimal support in S and give a basis for S loc = {f ∈ S: supp f ⊆ Ω} , with Ω a convex subset of R 2 .