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 formor 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.

Random sampling in shift invariant spaces

“…Our proof follows the line of [24]. For given ℓ ∈ N, we construct a 2 −ℓ -covering for V K (ϕ) with respect to ∥ · ∥ L ∞ (C K ) .…”

“…Using some methods taken from [24], we conclude that with overwhelming probability, the sampling inequality (1.1) holds for certain compact subsets of V (ϕ) when the sampling size is sufficiently large.…”

“…The general context of learning from random sampling has been studied by Cucker, Smale, Zhou et al (see [19][20][21][22]). Recently, the random sampling problems have been studied by Bass and Gröchenig in the multivariate trigonometric polynomial spaces [23] and bandlimited function spaces [24]. The purpose of this paper is to consider for some randomly chosen samples {x j ∈ X : j = 1, .…”

Relevant sampling in finitely generated shift-invariant spaces

“…There exist several well-understood ways of formulating these restrictions, eg. by assuming the analog signal f to be bandlimited, or more generally, that it belongs to a shift-invariant space [1,2,3,5,6,10,11,15,16,18,19,20]. Bandlimited signals of finite energy are completely characterized by their regular samples if they are taken at a sufficiently high rate (Nyquist criterion), as described by the famous classical Shannon sampling theorem.…”

“…The general context of learning from random sampling has been studied by Cucker, Smale, Zhou, et al (see [9,17]). Recently, the random sampling problems were studied by Bass and Gröchenig in the multivariate trigonometric polynomials spaces [4] and bandlimited functions spaces [5,6]. Yang and Wei discussed the problem when some randomly chosen samples X = {x j : j ∈ J} forms a set of sampling in the shift-invariant space [20].…”