Construction of positive definite cubature formulae and approximation of functions via Voronoi tessellations

Let ⊂ Rd be a compact convex set of positive measure. In a recent paper, we established a definiteness theory for cubature formulae of order two on . Here we study extremal properties of those positive definite formulae that can be generated by a centroidal Voronoi tessellation of . In this connection we come across a class of operators of the form Ln[ f ](x) := ∑n i=1 φi(x)( f (yi)+ 〈∇ f (yi), x − yi〉), where y1, . . . , yn are distinct points in and {φ1, . . . , φn} is a partition of unity on . We present best possible pointwise error estimates and describe operators Ln with a smallest constant in an Lp error estimate for 1 ≤ p < ∞. For a generalization, we introduce a new type of Voronoi tessellation in terms of a twice continuously differentiable and strictly convex function f . It allows us to describe a best operator Ln for approximating f by Ln[ f ] with respect to the Lp norm.