Cubature formulae and orthogonal polynomials
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. V: quadrature and orthogonal polynomials
Lower bound for the number of nodes of cubature formulae on the unit ball
Journal of Complexity
An encyclopaedia of cubature formulas
Journal of Complexity
Constructing cubature formulae of degree 5 with few points
Journal of Computational and Applied Mathematics
Hi-index | 7.29 |
In this note cubature formulae of degree 5 are studied for n-dimensional integrals over the ball with constant weight function. We apply the method of reproducing kernel to show that the existence of such formulae attaining the best known lower bound is equivalent to the existence of tight spherical 5-designs. The known results concerning spherical 5-designs show that the lower bound for the integral under consideration will not be attained in general. The bound will be attained for n = 2,3,7,23 and possibly for n = (2p+1)2-2, p ≥ 5. In all other cases the bound must be increased at least by 1, in particular, Stroud's formulae for n = 4,5,6,7 are minimal.