On generalized invariant cubature formulae

An important quality criterion of cubature formulae is their algebraic or trigonometric degree of exactness. The invariant theory is a powerful tool to construct cubature formulae of a given degree. In this paper, a quantitative expression is established for the classical invariant cubature formulas (ICFs). Motivated by this expression (or structure), we generalize the concept of ICFs and extend the famous Sobolevʹs Theorem on ICFs. The transformations allowed are no longer just orthogonal transformations. We illustrate the concepts and the constructions of the generalized ICFs by several examples.