Random sampling of bandlimited functions

Abstract

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 ∈ J ⊂ ℕ? We estimate the probability that a sampling inequality of the form

$$ A\left\| f \right\|_2^2 \leqslant \sum\limits_{j \in J} {|f(x_j )|^2 \leqslant B\left\| f \right\|_2^2 } $$

hold uniformly for all functions f ∈ L ²(ℝ^d) with supp $\hat f \subseteq [ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2},{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}]^d$ 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 ℝ^d. 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.