  In many applications, it is impractical -- if not even impossible -- to
obtain data to fit a known cubature formula (CF). Instead, experimental data is
often acquired at equidistant or even scattered locations. In this work, stable
(in the sense of nonnegative only cubature weights) high-order CFs are
developed for this purpose. These are based on the approach to allow the number
of data points N to be larger than the number of basis functions K which are
integrated exactly by the CF. This yields an (N-K)-dimensional affine linear
subspace from which cubature weights are selected that minimize certain norms
corresponding to stability of the CF. In the process, two novel classes of
stable high-order CFs are proposed and carefully investigated.
