Cubature Formulas of Infinite Order

We speak about a cubature formula of infinite order whenever the error of the formula is O(h m ) for all integer m and all functions in the space under study. Here h is the mesh-size of the lattice of integration. The Mean Value Theorem for harmonic functions provides the simplest example of a formula of such kind. In § 6 and § 8 of the current section we consider not so obvious examples of cubature and quadrature formulas of infinite order. The starting point of our study is the next observation. The estimate of accuracy of a formula of approximate integration by means of the norm of its error is unimprovable as long as we consider the whole math space. However, for every individual function φ in math the vanishing of (l h , φ) as h → 0, i.e., weak convergence, is always faster. This circumstance prompts us to seek for formulas of infinite order on various subspaces of math . As such we naturally take classes of infinitely differentiable functions, in particular, the Gevrey classes. The properties of the latter are scrutinized in § 2-§ 5 and § 7 of the current chapter. We thus address the formulas whose rate of convergence exceeds the rate we may expect from the formulas of a given polynomial degree. A similar phenomenon is referred to as superconvergence in [221].