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==Preliminaries==
[[Image:Hexagonal_sampling_lattice.png|thumb|Fig. 1: A hexagonal sampling lattice <math>\Lambda</math> and its generating vectors ''v''<sub>1</sub> and ''v''<sub>2</sub>|right|400px]]
[[Image:Reciprocal_lattice.png|thumb|Fig. 2: The reciprocal lattice <math>\Gamma</math> corresponding to the lattice <math>\Lambda</math> of Fig. 1 and its generating vectors ''u''<sub>1</sub> and ''u''<sub>2</sub>|right|400px]]
The concept of a [[Bandlimiting|bandlimited]] function in one dimension can be generalized to the notion of a wavenumber-limited function in higher dimensions. Recall that the [[Fourier transform]] of an integrable function ''ƒ(.)'' on ''n''-dimensional Euclidean space is defined as:
:<math>\hat{f}(\xi) = \mathcal{F}(f)(\xi) = \int_{\Re^n} f(x) e^{-2\pi i \langle x,\xi \rangle} \, dx</math>
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\Gamma = \left\{ \sum_{i=1}^n a_i u_i \; | \; a_i \in\Bbb{Z} \right\}
</math>
where the vectors <math>u_i</math> are chosen to satisfy <math>\langle u_i, v_j \rangle = 2\pi\delta_{ij}</math>. An example of a sampling lattice is depicted in Figure 1. The corresponding reciprocal lattice is shown in Figure 2.
==The theorem==
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