Sinc Function - Multidimensions

Multidimensions

The tensor-product of 1-D sinc functions readily provides a multivariate sinc function for the square, Cartesian, grid (Lattice): 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 (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, body centered cubic, face centered cubic and other higher dimensional lattices can be explicitly derived using the geometric properties of Brillouin zones and their connection to zonotopes.

For example, a hexagonal lattice can be generated by the (integer) Linear span of the vectors and . Denoting and, one can derive the sinc function for this hexagonal lattice as:

.

This construction can be used to design Lanczos window for general multidimensional lattices.