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 i(x)(f(yi) + f(yi), x - yi ), where y1, . . . , yn are distinct points in and {1, . . . , n} 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. Mathematics Subject Classification (2000): Primary 05B45, 41A63, 41A80, 52C22, 65D30, 65D32, Secondary 06A06, 26B2...

• Type: Journal
• Year: 2010
• Where: ADCM
• Authors: Allal Guessab, Gerhard Schmeisser
• Keywords: ADCM 2010, Centroidal Voronoi Tessellation, Definite Formulae, Voronoi Tessellation