Random Sampling of Bandlimited Functions

We consider the problem of random sampling for bandlimited functions. When can a bandlimited function f be recovered from randomly chosen samples f(xj), j ∈ J ⊂ N? We estimate the probability that a sampling inequality of the form A‖f‖ 2 ≤ ∑ j∈J |f(xj)| ≤ B‖f‖2 hold uniformly for all functions f ∈ L(R) with supp f̂ ⊆ [−1/2, 1/2] or for some subset of bandlimited functions. In contrast to discrete models, the space of bandlimited functions is infinitedimensional and its functions “live” on the unbounded set R. These facts raise 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 bandlimited functions and for sufficiently large sampling size.