On the number of nodes in \(n\)-dimensional cubature formulae of degree 5 for integrals over the ball
The paper deals with a problem associated with cubature formulas for the integral \(\int_B f(x)\,dx\), where \(B\) is the \(n\)-dimensional unit ball. Given the set of polynomials of degree \(m\) in \(n\) variables, it is an important question to find cubature formulas that are exact for all these polynomials. Particularly significant is the minimal formula, i.e.\ the formula with the smallest number of nodes. The knowledge about such minimal formulas is very limited at the moment. The paper under review gives some useful information about the minimal formulas in the case \(m=5\). The authors discuss classical lower bounds for the number of nodes of a minimal formula, show some cases when these bounds are sharp and improve the bounds in certain other cases, and they provide information about the location of the nodes in the cases when the classical bounds are attained. The proofs are based on methods using reproducing kernels and spherical 5-designs, and they provide additional useful information about spherical 5-designs.
scientific article; zbMATH DE number 2091408
scientific article