We consider a function 
in which the spatial domain
is a periodic extension of a unit cell. We shall first formally
describe this periodicity. Let M be a 
matrix which is
invertible and such that:
where 
is any D-dimensional integer vector (i.e., having integer
valued coordinates). Now, clearly, every point in space 
can be
written uniquely as:
where 
is a D-dimensional integer vector and 
is a vector
each coordinate of which satisfies 
. The unit cell 
is the region of space corresponding to all points 
. It can be shown that the volume of the unit cell is 
.
The set of all points 
of the form 
is
called the lattice induced by M. Basically, every point in space is
a point in the unit cell translated by a lattice vector. Note that the sum
of two lattice vectors is a lattice vector, and the periodicity of the
function f implies its value is invariant under translation by a lattice
vector. Define the matrix 
obtained by inverting and transposing M
as:
Then we can associate a lattice and unit cell with , called the reciprocal lattice 
and the reciprocal
unit cell 
, respectively. The reciprocal
unit cell is also called the Brillouin zone. If we consider wavenumber
space, each vector 
is written uniquely as:
where 
is a D-dimensional integer vector and 
has
all ordinates 
. The reciprocal lattice vectors are points of
the form 
.
The basic result is that the Fourier transform of a periodic function with
unit cell specified by M has a discrete spectrum located at points in the
reciprocal lattice specified by . That is, the wavenumber vectors
are constrained to lie at the reciprocal lattice points. The explicit
transform and inverse transform formulas are:
and:
We can interpret the discrete spectrum as a continuous spectrum consisting of Dirac impulse functions located at the reciprocal lattice points:
