Manuel Gräf

François Bachoc, Martin Ehler, Manuel Gräf

Motivated by the construction of confidence intervals in statistics, we study optimal configurations of $2^d-1$ lines in real projective space $RP^{d-1}$. For small $d$, we determine line sets that numerically minimize a wide variety of potential functions among all configurations of $2^d-1$ lines through the origin. Numerical experiments verify that our findings enable to assess efficiently the tightness of a bound arising from the statistical literature.

Martin Ehler, Manuel Gräf

The decomposition of polynomial spaces on unions of Grassmannians $\mathcal G_{{k_1},d}\cup\ldots\cup \mathcal G_{{k_r},d}$ into irreducible orthogonally invariant subspaces and their reproducing kernels are investigated. We also generalize the concepts of cubature points and $t$-designs from single Grassmannians to unions. We derive their characterization as minimizers of a suitable energy potential to enable $t$-design constructions by numerical optimization. We also present new analytic families of $t$-designs for $t=1,2,3$.