Seminar
ON THE EXISTENCE OF CUBATURE FORMULAS
by UJUE ETAYO (TU GRAZ)
Location: 200B.02.18, Leuven
Time: Tuesday October 22, 2019 at
14:00
We’re all familiar with the notion of quadrature formulas through which we can compute approximately the integral of a function on a given domain by computing the weighted sum of the function evaluated on a set of points of the domain. We call such a quadrature formula a cubature formula if the error is zero, i.e. if we can compute exactly the integral by the weighted sum. The existence of cubature formulas in general domains and for general spaces of functions is not trivial. Here we prove an asymptotic bound for the minimal number of points that we need to build this formulas in compact riemannian manifolds for perfect integration of diffusion polynomials. The proof follows the general idea presented by Bondarenko, Radchenko and Viazovska for the existence of t-designs. Nevertheless, new results have to be implemented, such as the existence of a certain kind of partitions on riemannian manifolds or the statement of some Marzinkievich-Zygmund inequalities for the gradients of the diffusion polynomials. The results here presented are a join work with B. Gariboldi, G. Gigante, M. Ehler and T. Peter.
