Near-minimal cubature formulae on the disk

Abstract

The construction of (near-)minimal cubature formulae on the disk is still a complicated subject on which many results have been published. We restrict ourselves to the case of radial weight functions and make use of a recent connection between cubature and the concept of multivariate spherical orthogonal polynomials to derive a new system of equations defining the nodes and weights of (near-)minimal rules for general degree m = 2n − 1, n ≥ 2. The approach encompasses all previous derivations. The new system is small and may consist of only (n + 1) 2 /4 equations when n is odd and n(n + 2)/4 equations when n is even. It is valid for general n and has a Prony-like structure. It may admit a unique solution (such as for n = 3) or an infinity of solutions (such as for n = 7). In Section 2, the new approach is described, whereas the new system is derived in Sections 3 and 4. All well-known (near-)minimal cubature rules can be reobtained. Some typical illustrations of how this works are given in Section 5. We expect that this unifying theory will shed new light on the topic of cubature, in particular with respect to the discovery of new bounds on the number of nodes and their connection with the zeros of multivariate orthogonal polynomials.

Construction and application of provable positive and exact cubature formulas

“…Moreover, we assume that (C N ) N ∈N is a CR with C N being positive and exact for all d-dimensional polynomials up to total degree m = m(N ). 7 That is, F K (Ω) = P m (R d ). In this case, the following result holds for the LS-CFs.…”

“…At the same time, it should be pointed out that Theorem 1.1 only provides an upper bound for the number of data points that are needed for a positive and F K (Ω)-exact CF. Indeed, for standard domains (e. g. Ω = [0, 1] d ) and weight functions (e. g. ω ≡ 1) as well as classical functions spaces (e. g. algebraic polynomials), it is possible to construct CFs that use even fewer points [75,73,45,11,10,7,115]. Such CFs are referred to as minimal or near-minimal CFs and usually utilize some kind of symmetry in the domain, weight function, and function space.…”