Random Sampling of Entire Functions of Exponential Type in Several Variables

We consider the problem of random sampling for band-limited functions. When can a band-limited function f be recovered from randomly chosen samples f(xj), j ∈ N? We estimate the probability that a sampling inequality of the form A‖f‖ 2 ≤ ∑ j∈N |f(xj)| ≤ B‖f‖2 hold uniformly all functions f ∈ L(R) with supp f̂ ⊆ [−1/2, 1/2] or some subset of band-limited functions. In contrast to discrete models, the space of band-limited functions is infinitedimensional and its functions “live” on the unbounded set R. This fact raises new problems and leads to both negative and positive results. (a) With probability one, the sampling inequality fails for any reasonable definition of a random set on R, e.g., for spatial Poisson processes or uniform distribution over disjoint cubes. (b) With overwhelming probability, the sampling inequality holds for certain compact subsets of the space of band-limited functions and for sufficiently large sampling size.