Minimal cubature formulae of degree 2k−1 for two classical functionals

Cubature formulae with the number of nodes equal to Möller's lower bound are rare. In this paper, the relation between real polynomial ideals and cubature formulae is used to construct such minimal formulae of arbitrary odd degree for two classical integrals. We found general expressions for bases of these ideals and closed formulae for almost all nodes. We proved that all nodes are inside the domain of integration.