Constructing cubature formulae by the method of reproducing kernel (Q1972276)

The method of reproducing kernel for constructing cubature formulae on the unit ball \(B^d\) and on the triangle was described. Compact formulae of the reproducing kernels studied by \textit{H. M. Möller} [Numer. Math. 25, 185-200 (1976; Zbl 0319.65019)] were used for the construction. At first the construction of a family of cubature formulae on \(B^d\) was discussed. Modification of the reproducing kernel method was given. The construction shows that the reproducing kernel method works for all \(n\) in every dimension. Different formulae from this method can be obtained. Cases of formulae of lower degree with nodes inside \(B^d\) and higher degree were dealt with. Cubature formulae on triangles hold for a family of weight functions and on a \(d\)-dimensional simplex. As an example the nodes and weights of the formulae of degrees 4 and 6 are tabulated.