ABSTRACT
In the previous section we provided a method of constructing cubature formulae of degree 2n – 1. For a given weight function there could be many such formulae with different numbers of nodes; among them, those with the smallest possible number we call minimal formulae. However, solving the nonlinear equations derived for the nodes of a formula is usually very difficult; the only practical way of knowing when a formula is minimal seems to be comparing its number of nodes with a known lower bound. The first lower bound is given in (1.2.9) which can be established very easily. Deeper results have been discovered, mainly by Mysovskikh and Möller. There are basically two lower bounds of general nature, both in their final form are due to Möller in his 1973 thesis. The first bound is stated in (1.2.10), the way it is proved also leads to a characterization of the cubature formulae which attain it. Although Möller’s characterization in [16–18] is not cleanly formulated and a little difficult to use; there have been some improvements (cf. [20, 23, 28]), most later papers have been centered around this lower bound and the method he used.
