In this paper, we investigate frames for L2 [- π, π]d consisting of exponential functions in connection to oversampling and nonuniform sampling of bandlimited functions. We derive a multidimensional nonuniform oversampling formula for bandlimited functions with a fairly general frequency domain. The stability of said formula under various perturbations in the sampled data is investigated, and a computationally manageable simplification of the main oversampling theorem is given. Also, a generalization of Kadec's 1/4 theorem to higher dimensions is considered. Finally, the developed techniques are used to approximate biorthogonal functions of particular exponential Riesz bases for L2 [- π, π], and a well-known theorem of Levinson is recovered as a corollary. © 2009 Elsevier Inc. All rights reserved.
Bailey, B. (2010). Sampling and recovery of multidimensional bandlimited functions via frames. Journal of Mathematical Analysis and Applications, 367(2), 374–388. https://doi.org/10.1016/j.jmaa.2009.12.051