Cubature formulae which are exact on spaces P, intermediate between \(P_ k\) and \(Q_ k\) (Q1070150)

We search, from the orthogonal polynomial theory, for conditions which allow to obtain cubature formulae on compacts of \(R^ n\), with weight function, and which are exact on the space \(R_{(k_ 1,k_ 2,\ldots,k_ n)}\) of all polynomials of degree \(\leq k_ i\) respectively to each variable \(x_ i\), \(1\leq i\leq n\).