Multivariate Gaussian cubature formulae
Gaussian quadrature can be briefly described as follows: There is a one- parameter family of minimal formulae of degree \(2n-2\) the nodes of which are the roots of \(p_ n+ \rho p_{n-1}\), \(\rho\neq 0\), where \(\{p_ n\}_{n=0}^ \infty\), denotes the system of orthogonal polynomials w.r.t. the integral under consideration; for \(\rho=0\) one even obtains a formula of degree \(2n-1\). In the multi-dimensional case, the structure of minimal cubature formulae is more complicated. For integrals on \(\mathbb{R}^ d\), \(d\geq 1\), \(\dim \prod_{[m/2]}^ d\) is a lower bound for the number of nodes of a formula of degree \(m\), where \(\prod_ m^ d\) is the linear space of all polynomials of total degree less than or equal to \(m\). As stated above, the bound is attained for \(d=1\) for arbitrary integrals; it is not, however, for classical integrals defined on the plane as has been observed by J. Radon in the late 40s. I. P. Mysovskikh and his students studied this problem intensively and came up with partial answers. Here we construct two classes of integrals on \(\mathbb{R}^ d\) allowing Gaussian cubature.
scientific article; zbMATH DE number 734520