The Second Stage: Finding the Weights

In the cases where the calculated nodes can be shown to be spherical designs (N=2,4,6,12), equal weights are optimal. This can be seen with the aid of Theorems 3 and 4. In all other cases appropriate weights have to be calculated.

With the aid of multivariate interpolation theory this is at least possible if the number N of nodes equals the dimension of the linear space of polynomials . For r=3 this means that .

Let and let be the symmetric matrix being composed of the evaluation of the kernel at the nodes , .

If are a fundamental system, i. e. , then the interpolation property
 
and the following consequence of the reproducing property of
 
are valid. Here,  denote the Lagrange fundamental polynomials. Equations (6) and (7) lead to

which means that the matrices and are inverse to each other.

Consider now a cubature formula

The function f can be substituted in the usual way by its interpolation polynomial p with regard to the nodes . Then

that is

Because and this yields

which is the row-sum of the matrix L.

Thus, because of (1), the weights of the cubature formula can be computed as the row-sum of the symmetric matrix

The Gegenbauer polynomials of degree m with index r/2 occuring in this composition have the representation

with .

They can be evaluated with the aid of their recurrence relation (see e. g. Reimer [16])
 

The inversion of the matrix can be avoided. With

we obtain the vector of weights as solution of the linear system of equations
 
In the following, we study the case r=3, that is , in detail, but the theory can be extended to higher dimensions without any difficulties.

The cases cannot be handled yet except for those cases in which the nodes are spherical designs. These are the cases N = 2,4,6,12 . The weights then can be chosen to be . The cases N=3,5,7 also show a regular behaviour, but up to now it could not be decided if these nodes are spherical designs. As interpolatory formulae our cubatures have a degree of exactness equal to m, whereas the spherical designs yield a degree of exactness equal to 2m or 2m+1, respectively.