Reproducing kernels for the irreducible components of polynomial spaces on unions of Grassmannians
Provides theoretical foundations and new constructions for t-designs on unions of Grassmannians, benefiting researchers in approximation theory and numerical analysis.
The paper decomposes polynomial spaces on unions of Grassmannians into irreducible subspaces and derives reproducing kernels, generalizing cubature points and t-designs to unions. New analytic families of t-designs for t=1,2,3 are presented.
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 $t=1,2,3$.