Dolomites Research Notes on Approximation

We show that for the d-cube K = [−1,1]d ⊂ Rd , there is for degree 2 a symmetric optimal design supported on the discrete set consisting of the vertices, the edge midpoints and the origin with cardinality 2d + d2d−1 + 1. In general there is a continuum of possible optimal designs with, however, a support of larger cardinality. We also consider numerically the degree three case for the square [−1,1] ⊂ R2. Our calculations indicate that there is an optimal measure supported on 16 points but that these do not form a regular grid.