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(x(j)), j is an element of J subset of N? We estimate the probability that a sampling inequality of the form A parallel to f parallel to(2)(2) <= Sigma j is an element of J vertical bar f(x(j))vertical bar(2) <= B parallel to f parallel to(2)(2) hold uniformly for all functions f is an element of L-2 (R-d) with supp (f) over cap subset of [-1/ 2, 1/2](d) or for some subset of bandlimited functions. In contrast to discrete models, the space of bandlimited functions is infinite- dimensional and its functions live on the unbounded set Rd. 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-d, 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.