Random Sampling of Entire Functions of Exponential Type in Several Variables

DOI10.1007/S11856-010-0036-7arXiv0706.3818MaRDI QIDQ6206015

Richard F. Bass, Karlheinz Gröchenig

Publication date: 26 June 2007

Abstract: We consider the problem of random sampling for band-limited functions. When can a band-limited function $f$ be recovered from randomly chosen samples $f(x_j),jinmathbbN$? We estimate the probability that a sampling inequality of the form A|f|_2^2 leq sum_{jin mathbb{N}} |f(x_j)|^2 leq B |f|_2^2 hold uniformly all functions $fin{L}^2(mathbb{R}^d)$ with supp $hatfsubseteq[-1/2,1/2{]}^d$ or some subset of �dl functions. In contrast to discrete models, the space of band-limited functions is infinite-dimensional and its functions "live" on the unbounded set $mathbb{R}^d$. 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 $mathbb{R}^d$, 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 band-limited functions and for sufficiently large sampling size.

Mathematics Subject Classification ID

Inference from spatial processes (62M30) Inequalities; stochastic orderings (60E15) General harmonic expansions, frames (42C15)

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