International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 4344

Let Λ denote the nonstandard lattice consisting of all vectors of the form , where the are rational integers and are n linearly independent vectors in . Let R be the corresponding lattice distribution: .
Let A be the nonsingular matrix whose successive columns are the coordinates of vectors in the standard basis of ; A will be called the period matrix of Λ, and the mapping will be denoted by A. According to Section 1.3.2.3.9.5 we havefor any , and hence . By Fourier transformation, according to Section 1.3.2.5.5,which we write:with
is a lattice distribution:associated with the reciprocal lattice whose basis vectors are the columns of . Since the latter matrix is equal to the adjoint matrix (i.e. the matrix of cofactors) of A divided by det A, the components of the reciprocal basis vectors can be written down explicitly (see Section 1.3.4.2.1.1 for the crystallographic case ).
A distribution T will be called Λperiodic if for all ; as previously, T may be written for some motif distribution with compact support. By Fourier transformation,so that is a weighted reciprocallattice distribution, the weight attached to node being times the value of the Fourier transform of the motif .
This result may be further simplified if T and its motif are referred to the standard period lattice by defining t and so that , , . Thenhenceso thatin nonstandard coordinates, whilein standard coordinates.
The reciprocity theorem may then be written:in nonstandard coordinates, or equivalently:in standard coordinates. It gives an ndimensional Fourier series representation for any periodic distribution over . The convergence of such series in will be examined in Section 1.3.2.6.10.