Generalized Regular Sampling of Trigonometric Polynomials and Optimal Sensor Arrangement

We address the
optimal sensor arrangement problem
, which is the determination of a geometric configuration of sensors such that the mean-squared error (MSE) in the estimation of an unknown trigonometric polynomial is minimum. Unsurprisingly, an arrangement in which sensors are spaced uniformly in each dimension is optimal. However, for multidimensional problems the minimum MSE is achieved with a much larger class of configurations that we call
generalized regular arrangements
. These arrangements are not necessarily generated by lattices and may exhibit great nonuniformity locally.