Sampling and recovery of multidimensional bandlimited functions via frames

In this paper, we investigate frames for L2d[−π,π] 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.