Multidimensional sampling: Difference between revisions

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==Preliminaries==

[[Image:Hexagonal_sampling_lattice.png|thumb|Fig. 1: A hexagonal sampling lattice <math>\Lambda</math> and its generatingbasis vectors ''v''₁ and ''v''₂|right|200px]]

[[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 generatingbasis vectors ''u''₁ and ''u''₂ (figure not to scale).|right|200px]]

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>