Editing Sinc function (section)

== Higher dimensions ==
The product of 1-D sinc functions readily provides a [[multivariable calculus|multivariate]] sinc function for the square Cartesian grid ([[Lattice graph|lattice]]): {{math|sinc<sub>C</sub>(''x'', ''y'') {{=}} sinc(''x'') sinc(''y'')}}, whose [[Fourier transform]] is the [[indicator function]] of a square in the frequency space (i.e., the brick wall defined in 2-D space). The sinc function for a non-Cartesian [[Lattice (group)|lattice]] (e.g., [[hexagonal lattice]]) is a function whose [[Fourier transform]] is the [[indicator function]] of the [[Brillouin zone]] of that lattice. For example, the sinc function for the hexagonal lattice is a function whose [[Fourier transform]] is the [[indicator function]] of the unit hexagon in the frequency space. For a non-Cartesian lattice this function can not be obtained by a simple tensor product. However, the explicit formula for the sinc function for the [[hexagonal lattice|hexagonal]], [[body-centered cubic]], [[face-centered cubic]] and other higher-dimensional lattices can be explicitly derived<ref name="multiD">{{cite journal |last1=Ye |first1= W. |last2=Entezari |first2= A. |title=A Geometric Construction of Multivariate Sinc Functions |journal=IEEE Transactions on Image Processing |volume=21 |issue=6 |pages=2969–2979 |date=June 2012 |doi=10.1109/TIP.2011.2162421 |pmid=21775264 |bibcode=2012ITIP...21.2969Y|s2cid= 15313688 }}</ref> using the geometric properties of Brillouin zones and their connection to [[zonohedron|zonotopes]].

For example, a [[hexagonal lattice]] can be generated by the (integer) [[linear span]] of the vectors
<math display="block">
  \mathbf{u}_1 = \begin{bmatrix} \frac{1}{2} \\  \frac{\sqrt{3}}{2} \end{bmatrix} \quad \text{and} \quad
  \mathbf{u}_2 = \begin{bmatrix} \frac{1}{2} \\ -\frac{\sqrt{3}}{2} \end{bmatrix}.
</math>

Denoting
<math display="block">
  \boldsymbol{\xi}_1 =  \tfrac{2}{3} \mathbf{u}_1, \quad
  \boldsymbol{\xi}_2 =  \tfrac{2}{3} \mathbf{u}_2, \quad
  \boldsymbol{\xi}_3 = -\tfrac{2}{3} (\mathbf{u}_1 + \mathbf{u}_2), \quad
          \mathbf{x} = \begin{bmatrix} x \\ y\end{bmatrix},
</math>
one can derive<ref name="multiD" /> the sinc function for this hexagonal lattice as
<math display="block">\begin{align}
  \operatorname{sinc}_\text{H}(\mathbf{x}) = \tfrac{1}{3} \big(
    &      \cos\left(\pi\boldsymbol{\xi}_1\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_2\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_3\cdot\mathbf{x}\right) \\
    & {} + \cos\left(\pi\boldsymbol{\xi}_2\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_3\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_1\cdot\mathbf{x}\right) \\
    & {} + \cos\left(\pi\boldsymbol{\xi}_3\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_1\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_2\cdot\mathbf{x}\right)
  \big).
\end{align}</math>

This construction can be used to design [[Lanczos window]] for general multidimensional lattices.<ref name="multiD" />