Lower bound for the number of nodes of cubature formulae on the unit ball

Abstract

A lower bound for positive cubature formulae on the unit ball is given. For the Chebyshev weight function on the ball in ℝ2, the new bound shows that a positive cubature formula of degree s with all nodes inside the ball will need at least Ns ≥ 0.13622s2 (1 + O(s-1)) number of nodes, in comparison with the classical lower bound of Ns ≥0.125s2 (1 + O (s-1)). © 2003 Elsevier Science (USA). All rights reserved.